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Límite de (-sin(3*x)+tan(3*x))/(2*x^2)
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Límite de (-1+sqrt(1+x^2))/(-4+sqrt(16+x^2))
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Expresiones idénticas
- (tres +sqrt(x^ dos - seis *x)-x)/(tres +x-sqrt(x^ dos + seis *x))
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- (tres más raíz cuadrada de (x en el grado dos menos seis multiplicar por x) menos x) dividir por (tres más x menos raíz cuadrada de (x en el grado dos más seis multiplicar por x))
- (3+√(x^2-6*x)-x)/(3+x-√(x^2+6*x))
- (3+sqrt(x2-6*x)-x)/(3+x-sqrt(x2+6*x))
- 3+sqrtx2-6*x-x/3+x-sqrtx2+6*x
- (3+sqrt(x²-6*x)-x)/(3+x-sqrt(x²+6*x))
- (3+sqrt(x en el grado 2-6*x)-x)/(3+x-sqrt(x en el grado 2+6*x))
- (3+sqrt(x^2-6x)-x)/(3+x-sqrt(x^2+6x))
- (3+sqrt(x2-6x)-x)/(3+x-sqrt(x2+6x))
- 3+sqrtx2-6x-x/3+x-sqrtx2+6x
- 3+sqrtx^2-6x-x/3+x-sqrtx^2+6x
- (3+sqrt(x^2-6*x)-x) dividir por (3+x-sqrt(x^2+6*x))
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Expresiones semejantes
- (3+sqrt(x^2-6*x)-x)/(3-x-sqrt(x^2+6*x))
- (3+sqrt(x^2-6*x)+x)/(3+x-sqrt(x^2+6*x))
- (3+sqrt(x^2-6*x)-x)/(3+x-sqrt(x^2-6*x))
- (3+sqrt(x^2+6*x)-x)/(3+x-sqrt(x^2+6*x))
- (3+sqrt(x^2-6*x)-x)/(3+x+sqrt(x^2+6*x))
- (3-sqrt(x^2-6*x)-x)/(3+x-sqrt(x^2+6*x))
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Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(3+2*x)-sqrt(-7+2*x)
- sqrt(-2+x)/(-4+x^2)
- sqrt(1+x+x^2)-sqrt(-1+x^2-x)
- sqrt(1+x+x^2)-x
- sqrt(x)*(sqrt(1+x)-sqrt(x))
- Raíz cuadrada sqrt
- sqrt(3+2*x)-sqrt(-7+2*x)
- sqrt(-2+x)/(-4+x^2)
- sqrt(1+x+x^2)-sqrt(-1+x^2-x)
- sqrt(1+x+x^2)-x
- sqrt(x)*(sqrt(1+x)-sqrt(x))
Límite de la función (3+sqrt(x^2-6*x)-x)/(3+x-sqrt(x^2+6*x))
Solución
Ha introducido
[src]
/ __________ \
| / 2 |
|3 + \/ x - 6*x - x|
lim |---------------------|
x->oo| __________|
| / 2 |
\3 + x - \/ x + 6*x /
$$\lim_{x \to \infty}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right)$$
Limit((3 + sqrt(x^2 - 6*x) - x)/(3 + x - sqrt(x^2 + 6*x)), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty} \frac{1}{x - \sqrt{x^{2} + 6 x} + 3} = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{- x + \sqrt{x^{2} - 6 x} + 3} = -\infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{- x + \sqrt{x \left(x - 6\right)} + 3}{x - \sqrt{x \left(x + 6\right)} + 3}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{x - \sqrt{x^{2} + 6 x} + 3}}{\frac{d}{d x} \frac{1}{- x + \sqrt{x^{2} - 6 x} + 3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{1}{\left(- \frac{x}{\sqrt{x^{2} - 6 x}} + 1 + \frac{3}{\sqrt{x^{2} - 6 x}}\right) \left(\frac{2 x^{2}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} - \frac{2 x \sqrt{x^{2} + 6 x}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{12 x}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} - \frac{6 \sqrt{x^{2} + 6 x}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{9}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{x}{\sqrt{x^{2} - 6 x}} + 1 + \frac{3}{\sqrt{x^{2} - 6 x}}}}{\frac{d}{d x} \left(\frac{2 x^{2}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} - \frac{2 x \sqrt{x^{2} + 6 x}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{12 x}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} - \frac{6 \sqrt{x^{2} + 6 x}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{9}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{x \left(3 - x\right)}{\left(x^{2} - 6 x\right)^{\frac{3}{2}}} - \frac{3 \left(3 - x\right)}{\left(x^{2} - 6 x\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{x^{2} - 6 x}}}{\left(- \frac{x}{\sqrt{x^{2} - 6 x}} + 1 + \frac{3}{\sqrt{x^{2} - 6 x}}\right)^{2} \left(\frac{2 x^{2} \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} - \frac{2 x \left(x + 3\right)}{\sqrt{x^{2} + 6 x} \left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)} - \frac{2 x \sqrt{x^{2} + 6 x} \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} + \frac{4 x}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{12 x \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} - \frac{6 \left(x + 3\right)}{\sqrt{x^{2} + 6 x} \left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)} - \frac{2 \sqrt{x^{2} + 6 x}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} - \frac{6 \sqrt{x^{2} + 6 x} \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} + \frac{12}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{9 \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{x \left(3 - x\right)}{\left(x^{2} - 6 x\right)^{\frac{3}{2}}} - \frac{3 \left(3 - x\right)}{\left(x^{2} - 6 x\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{x^{2} - 6 x}}}{\left(- \frac{x}{\sqrt{x^{2} - 6 x}} + 1 + \frac{3}{\sqrt{x^{2} - 6 x}}\right)^{2} \left(\frac{2 x^{2} \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} - \frac{2 x \left(x + 3\right)}{\sqrt{x^{2} + 6 x} \left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)} - \frac{2 x \sqrt{x^{2} + 6 x} \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} + \frac{4 x}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{12 x \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} - \frac{6 \left(x + 3\right)}{\sqrt{x^{2} + 6 x} \left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)} - \frac{2 \sqrt{x^{2} + 6 x}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} - \frac{6 \sqrt{x^{2} + 6 x} \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} + \frac{12}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{9 \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}}\right)}\right)$$
=
$$-1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
oo/-oo,
tal que el límite para el numerador es
$$\lim_{x \to \infty} \frac{1}{x - \sqrt{x^{2} + 6 x} + 3} = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{- x + \sqrt{x^{2} - 6 x} + 3} = -\infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{- x + \sqrt{x \left(x - 6\right)} + 3}{x - \sqrt{x \left(x + 6\right)} + 3}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{x - \sqrt{x^{2} + 6 x} + 3}}{\frac{d}{d x} \frac{1}{- x + \sqrt{x^{2} - 6 x} + 3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{1}{\left(- \frac{x}{\sqrt{x^{2} - 6 x}} + 1 + \frac{3}{\sqrt{x^{2} - 6 x}}\right) \left(\frac{2 x^{2}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} - \frac{2 x \sqrt{x^{2} + 6 x}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{12 x}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} - \frac{6 \sqrt{x^{2} + 6 x}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{9}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{x}{\sqrt{x^{2} - 6 x}} + 1 + \frac{3}{\sqrt{x^{2} - 6 x}}}}{\frac{d}{d x} \left(\frac{2 x^{2}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} - \frac{2 x \sqrt{x^{2} + 6 x}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{12 x}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} - \frac{6 \sqrt{x^{2} + 6 x}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{9}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{x \left(3 - x\right)}{\left(x^{2} - 6 x\right)^{\frac{3}{2}}} - \frac{3 \left(3 - x\right)}{\left(x^{2} - 6 x\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{x^{2} - 6 x}}}{\left(- \frac{x}{\sqrt{x^{2} - 6 x}} + 1 + \frac{3}{\sqrt{x^{2} - 6 x}}\right)^{2} \left(\frac{2 x^{2} \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} - \frac{2 x \left(x + 3\right)}{\sqrt{x^{2} + 6 x} \left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)} - \frac{2 x \sqrt{x^{2} + 6 x} \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} + \frac{4 x}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{12 x \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} - \frac{6 \left(x + 3\right)}{\sqrt{x^{2} + 6 x} \left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)} - \frac{2 \sqrt{x^{2} + 6 x}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} - \frac{6 \sqrt{x^{2} + 6 x} \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} + \frac{12}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{9 \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{x \left(3 - x\right)}{\left(x^{2} - 6 x\right)^{\frac{3}{2}}} - \frac{3 \left(3 - x\right)}{\left(x^{2} - 6 x\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{x^{2} - 6 x}}}{\left(- \frac{x}{\sqrt{x^{2} - 6 x}} + 1 + \frac{3}{\sqrt{x^{2} - 6 x}}\right)^{2} \left(\frac{2 x^{2} \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} - \frac{2 x \left(x + 3\right)}{\sqrt{x^{2} + 6 x} \left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)} - \frac{2 x \sqrt{x^{2} + 6 x} \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} + \frac{4 x}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{12 x \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} - \frac{6 \left(x + 3\right)}{\sqrt{x^{2} + 6 x} \left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)} - \frac{2 \sqrt{x^{2} + 6 x}}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} - \frac{6 \sqrt{x^{2} + 6 x} \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}} + \frac{12}{\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}} + \frac{9 \left(- \frac{2 x^{3} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + \frac{27 x \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{2 x \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} + \frac{4 x \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} + 4 x + \frac{12 x}{\sqrt{x^{2} + 6 x}} - \frac{18 \left(- x - 3\right) \sqrt{x^{2} - 6 x}}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} - \frac{27 \left(- x - 3\right)}{\left(x^{2} + 6 x\right)^{\frac{3}{2}}} + \frac{6 \left(x - 3\right)}{\sqrt{x^{2} - 6 x}} - \frac{18 \left(x - 3\right)}{\sqrt{x^{2} - 6 x} \sqrt{x^{2} + 6 x}} - 2 \sqrt{x^{2} - 6 x} - 12 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)}{\left(\frac{2 x^{3}}{\sqrt{x^{2} + 6 x}} - \frac{2 x^{2} \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 2 x^{2} - \frac{6 x^{2}}{\sqrt{x^{2} + 6 x}} + 2 x \sqrt{x^{2} - 6 x} + 12 x - \frac{27 x}{\sqrt{x^{2} + 6 x}} - 6 \sqrt{x^{2} - 6 x} + \frac{18 \sqrt{x^{2} - 6 x}}{\sqrt{x^{2} + 6 x}} - 9 + \frac{27}{\sqrt{x^{2} + 6 x}}\right)^{2}}\right)}\right)$$
=
$$-1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right) = -1$$
$$\lim_{x \to 0^-}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right) = 1$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right) = - \frac{2 + \sqrt{5} i}{-4 + \sqrt{7}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right) = - \frac{2 + \sqrt{5} i}{-4 + \sqrt{7}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right) = -1$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right) = 1$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right) = - \frac{2 + \sqrt{5} i}{-4 + \sqrt{7}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right) = - \frac{2 + \sqrt{5} i}{-4 + \sqrt{7}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- x + \left(\sqrt{x^{2} - 6 x} + 3\right)}{\left(x + 3\right) - \sqrt{x^{2} + 6 x}}\right) = -1$$
Más detalles con x→-oo