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- Límite de la función:
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Límite de -6+8*x/3
- Límite de (-2+x+x^2)/(2+x^2-3*x)
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Límite de x^(1/log(-1+e^x))
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Límite de x^(1-x)
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Expresiones idénticas
- (-sqrt(uno -sqrt(x))+cos(sin(x/ dos)))/x^ cuatro
- ( menos raíz cuadrada de (1 menos raíz cuadrada de (x)) más coseno de ( seno de (x dividir por 2))) dividir por x en el grado 4
- ( menos raíz cuadrada de (uno menos raíz cuadrada de (x)) más coseno de ( seno de (x dividir por dos))) dividir por x en el grado cuatro
- (-√(1-√(x))+cos(sin(x/2)))/x^4
- (-sqrt(1-sqrt(x))+cos(sin(x/2)))/x4
- -sqrt1-sqrtx+cossinx/2/x4
- (-sqrt(1-sqrt(x))+cos(sin(x/2)))/x⁴
- -sqrt1-sqrtx+cossinx/2/x^4
- (-sqrt(1-sqrt(x))+cos(sin(x dividir por 2))) dividir por x^4
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Expresiones semejantes
- (-sqrt(1+sqrt(x))+cos(sin(x/2)))/x^4
- (sqrt(1-sqrt(x))+cos(sin(x/2)))/x^4
- (-sqrt(1-sqrt(x))-cos(sin(x/2)))/x^4
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Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(3+x^2+8*x)-sqrt(3+x^2+4*x)
- sqrt(1+9*x^2)-3*x
- sqrt(3+x^2+2*x)-x
- sqrt(x+sqrt(x+sqrt(x)))/sqrt(1+x)
- sqrt(x)-log(x)
- Raíz cuadrada sqrt
- sqrt(3+x^2+8*x)-sqrt(3+x^2+4*x)
- sqrt(1+9*x^2)-3*x
- sqrt(3+x^2+2*x)-x
- sqrt(x+sqrt(x+sqrt(x)))/sqrt(1+x)
- sqrt(x)-log(x)
- Coseno cos
- cos(pi*x)^(sin(pi*x)/x)
- cos(2*x)^tan(x/2)
- cos(2*pi/x)/(-1+3*x)
- cos(2*n)/n
- cos(4/x)
- Seno sin
- sin(x)^(x^2)
- sin(x)/tan(4*x)
- sin(7*x)/tan(8*x)
- sin(7*x)/x^2
- sin(a)
Límite de la función (-sqrt(1-sqrt(x))+cos(sin(x/2)))/x^4
Solución
Ha introducido
[src]
/ ___________ \
| / ___ / /x\\|
|- \/ 1 - \/ x + cos|sin|-|||
| \ \2//|
lim |------------------------------|
x->0+| 4 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right)$$
Limit((-sqrt(1 - sqrt(x)) + cos(sin(x/2)))/x^4, x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} x^{4} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}\right)}{\frac{d}{d x} x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos{\left(\frac{x}{2} \right)}}{2} + \frac{1}{4 \sqrt{x} \sqrt{1 - \sqrt{x}}}}{4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos{\left(\frac{x}{2} \right)}}{2} + \frac{1}{4 \sqrt{x} \sqrt{1 - \sqrt{x}}}\right)}{\frac{d}{d x} 4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{4} - \frac{\cos^{2}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{4} + \frac{1}{- 16 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}} + 16 x \sqrt{1 - \sqrt{x}}} - \frac{1}{8 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}}{12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{4} - \frac{\cos^{2}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{4} + \frac{1}{- 16 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}} + 16 x \sqrt{1 - \sqrt{x}}} - \frac{1}{8 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}\right)}{\frac{d}{d x} 12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{24 \sqrt{x} \sqrt{1 - \sqrt{x}}}{- 256 x^{\frac{7}{2}} - 768 x^{\frac{5}{2}} + 768 x^{3} + 256 x^{2}} + \frac{4 \sqrt{x}}{- 256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 768 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 768 x^{3} \sqrt{1 - \sqrt{x}} + 256 x^{2} \sqrt{1 - \sqrt{x}}} - \frac{4 x}{- 256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 768 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 768 x^{3} \sqrt{1 - \sqrt{x}} + 256 x^{2} \sqrt{1 - \sqrt{x}}} - \frac{16 \sqrt{1 - \sqrt{x}}}{- 256 x^{\frac{7}{2}} - 768 x^{\frac{5}{2}} + 768 x^{3} + 256 x^{2}} + \frac{3 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{8} + \frac{\sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos^{3}{\left(\frac{x}{2} \right)}}{8} + \frac{\sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos{\left(\frac{x}{2} \right)}}{8} - \frac{1}{- 32 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 32 x^{2} \sqrt{1 - \sqrt{x}}} + \frac{3}{16 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}}{24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{24 \sqrt{x} \sqrt{1 - \sqrt{x}}}{- 256 x^{\frac{7}{2}} - 768 x^{\frac{5}{2}} + 768 x^{3} + 256 x^{2}} + \frac{4 \sqrt{x}}{- 256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 768 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 768 x^{3} \sqrt{1 - \sqrt{x}} + 256 x^{2} \sqrt{1 - \sqrt{x}}} - \frac{4 x}{- 256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 768 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 768 x^{3} \sqrt{1 - \sqrt{x}} + 256 x^{2} \sqrt{1 - \sqrt{x}}} - \frac{16 \sqrt{1 - \sqrt{x}}}{- 256 x^{\frac{7}{2}} - 768 x^{\frac{5}{2}} + 768 x^{3} + 256 x^{2}} + \frac{3 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{8} + \frac{\sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos^{3}{\left(\frac{x}{2} \right)}}{8} + \frac{\sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos{\left(\frac{x}{2} \right)}}{8} - \frac{1}{- 32 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 32 x^{2} \sqrt{1 - \sqrt{x}}} + \frac{3}{16 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}\right)}{\frac{d}{d x} 24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{448 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} - \frac{128 x^{\frac{7}{2}}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} - \frac{704 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}{- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}} - \frac{8704 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}\right)} - \frac{128 x^{\frac{5}{2}}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} - \frac{256 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} - \frac{1792 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}{- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}} - \frac{10 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 1024 x^{\frac{11}{2}} - 3072 x^{\frac{9}{2}} + 3072 x^{5} + 1024 x^{4}\right)} - \frac{x^{\frac{3}{2}}}{- 3072 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 9216 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 9216 x^{5} \sqrt{1 - \sqrt{x}} + 3072 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{32 x^{4}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{1600 x^{3} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} + \frac{896 x^{3} \sqrt{1 - \sqrt{x}}}{- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}} + \frac{64 x^{3}}{- 65536 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 2293760 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 458752 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 458752 x^{7} \sqrt{1 - \sqrt{x}} + 2293760 x^{6} \sqrt{1 - \sqrt{x}} + 1376256 x^{5} \sqrt{1 - \sqrt{x}} + 65536 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{1216 x^{2} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} + \frac{3456 x^{2} \sqrt{1 - \sqrt{x}}}{- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}} + \frac{32 x^{2}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{x^{2}}{- 3072 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 9216 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 9216 x^{5} \sqrt{1 - \sqrt{x}} + 3072 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{1024 x \sqrt{1 - \sqrt{x}}}{3 \left(- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}\right)} + \frac{8 x \sqrt{1 - \sqrt{x}}}{3 \left(- 1024 x^{\frac{11}{2}} - 3072 x^{\frac{9}{2}} + 3072 x^{5} + 1024 x^{4}\right)} + \frac{\sqrt{1 - \sqrt{x}}}{1536 x^{\frac{7}{2}} + 512 x^{\frac{5}{2}} - 512 x^{4} - 1536 x^{3}} - \frac{\sin^{2}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{128} - \frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos^{2}{\left(\frac{x}{2} \right)}}{64} - \frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{384} + \frac{\cos^{4}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{384} + \frac{\cos^{2}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{96} + \frac{1}{9216 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} + 3072 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} - 3072 x^{4} \sqrt{1 - \sqrt{x}} - 9216 x^{3} \sqrt{1 - \sqrt{x}}} + \frac{1}{4608 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} + 1536 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} - 1536 x^{4} \sqrt{1 - \sqrt{x}} - 4608 x^{3} \sqrt{1 - \sqrt{x}}} - \frac{1}{6 \left(- 256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 768 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 768 x^{3} \sqrt{1 - \sqrt{x}} + 256 x^{2} \sqrt{1 - \sqrt{x}}\right)} - \frac{1}{- 1024 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 3072 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 3072 x^{3} \sqrt{1 - \sqrt{x}} + 1024 x^{2} \sqrt{1 - \sqrt{x}}} + \frac{1}{- 512 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} + 512 x^{3} \sqrt{1 - \sqrt{x}}} - \frac{5}{256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{448 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} - \frac{128 x^{\frac{7}{2}}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} - \frac{704 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}{- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}} - \frac{8704 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}\right)} - \frac{128 x^{\frac{5}{2}}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} - \frac{256 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} - \frac{1792 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}{- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}} - \frac{10 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 1024 x^{\frac{11}{2}} - 3072 x^{\frac{9}{2}} + 3072 x^{5} + 1024 x^{4}\right)} - \frac{x^{\frac{3}{2}}}{- 3072 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 9216 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 9216 x^{5} \sqrt{1 - \sqrt{x}} + 3072 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{32 x^{4}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{1600 x^{3} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} + \frac{896 x^{3} \sqrt{1 - \sqrt{x}}}{- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}} + \frac{64 x^{3}}{- 65536 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 2293760 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 458752 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 458752 x^{7} \sqrt{1 - \sqrt{x}} + 2293760 x^{6} \sqrt{1 - \sqrt{x}} + 1376256 x^{5} \sqrt{1 - \sqrt{x}} + 65536 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{1216 x^{2} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} + \frac{3456 x^{2} \sqrt{1 - \sqrt{x}}}{- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}} + \frac{32 x^{2}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{x^{2}}{- 3072 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 9216 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 9216 x^{5} \sqrt{1 - \sqrt{x}} + 3072 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{1024 x \sqrt{1 - \sqrt{x}}}{3 \left(- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}\right)} + \frac{8 x \sqrt{1 - \sqrt{x}}}{3 \left(- 1024 x^{\frac{11}{2}} - 3072 x^{\frac{9}{2}} + 3072 x^{5} + 1024 x^{4}\right)} + \frac{\sqrt{1 - \sqrt{x}}}{1536 x^{\frac{7}{2}} + 512 x^{\frac{5}{2}} - 512 x^{4} - 1536 x^{3}} - \frac{\sin^{2}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{128} - \frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos^{2}{\left(\frac{x}{2} \right)}}{64} - \frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{384} + \frac{\cos^{4}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{384} + \frac{\cos^{2}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{96} + \frac{1}{9216 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} + 3072 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} - 3072 x^{4} \sqrt{1 - \sqrt{x}} - 9216 x^{3} \sqrt{1 - \sqrt{x}}} + \frac{1}{4608 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} + 1536 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} - 1536 x^{4} \sqrt{1 - \sqrt{x}} - 4608 x^{3} \sqrt{1 - \sqrt{x}}} - \frac{1}{6 \left(- 256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 768 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 768 x^{3} \sqrt{1 - \sqrt{x}} + 256 x^{2} \sqrt{1 - \sqrt{x}}\right)} - \frac{1}{- 1024 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 3072 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 3072 x^{3} \sqrt{1 - \sqrt{x}} + 1024 x^{2} \sqrt{1 - \sqrt{x}}} + \frac{1}{- 512 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} + 512 x^{3} \sqrt{1 - \sqrt{x}}} - \frac{5}{256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}}}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} x^{4} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}\right)}{\frac{d}{d x} x^{4}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos{\left(\frac{x}{2} \right)}}{2} + \frac{1}{4 \sqrt{x} \sqrt{1 - \sqrt{x}}}}{4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos{\left(\frac{x}{2} \right)}}{2} + \frac{1}{4 \sqrt{x} \sqrt{1 - \sqrt{x}}}\right)}{\frac{d}{d x} 4 x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{4} - \frac{\cos^{2}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{4} + \frac{1}{- 16 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}} + 16 x \sqrt{1 - \sqrt{x}}} - \frac{1}{8 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}}{12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{4} - \frac{\cos^{2}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{4} + \frac{1}{- 16 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}} + 16 x \sqrt{1 - \sqrt{x}}} - \frac{1}{8 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}\right)}{\frac{d}{d x} 12 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{24 \sqrt{x} \sqrt{1 - \sqrt{x}}}{- 256 x^{\frac{7}{2}} - 768 x^{\frac{5}{2}} + 768 x^{3} + 256 x^{2}} + \frac{4 \sqrt{x}}{- 256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 768 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 768 x^{3} \sqrt{1 - \sqrt{x}} + 256 x^{2} \sqrt{1 - \sqrt{x}}} - \frac{4 x}{- 256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 768 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 768 x^{3} \sqrt{1 - \sqrt{x}} + 256 x^{2} \sqrt{1 - \sqrt{x}}} - \frac{16 \sqrt{1 - \sqrt{x}}}{- 256 x^{\frac{7}{2}} - 768 x^{\frac{5}{2}} + 768 x^{3} + 256 x^{2}} + \frac{3 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{8} + \frac{\sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos^{3}{\left(\frac{x}{2} \right)}}{8} + \frac{\sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos{\left(\frac{x}{2} \right)}}{8} - \frac{1}{- 32 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 32 x^{2} \sqrt{1 - \sqrt{x}}} + \frac{3}{16 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}}{24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{24 \sqrt{x} \sqrt{1 - \sqrt{x}}}{- 256 x^{\frac{7}{2}} - 768 x^{\frac{5}{2}} + 768 x^{3} + 256 x^{2}} + \frac{4 \sqrt{x}}{- 256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 768 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 768 x^{3} \sqrt{1 - \sqrt{x}} + 256 x^{2} \sqrt{1 - \sqrt{x}}} - \frac{4 x}{- 256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 768 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 768 x^{3} \sqrt{1 - \sqrt{x}} + 256 x^{2} \sqrt{1 - \sqrt{x}}} - \frac{16 \sqrt{1 - \sqrt{x}}}{- 256 x^{\frac{7}{2}} - 768 x^{\frac{5}{2}} + 768 x^{3} + 256 x^{2}} + \frac{3 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{8} + \frac{\sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos^{3}{\left(\frac{x}{2} \right)}}{8} + \frac{\sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos{\left(\frac{x}{2} \right)}}{8} - \frac{1}{- 32 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 32 x^{2} \sqrt{1 - \sqrt{x}}} + \frac{3}{16 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}\right)}{\frac{d}{d x} 24 x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{448 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} - \frac{128 x^{\frac{7}{2}}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} - \frac{704 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}{- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}} - \frac{8704 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}\right)} - \frac{128 x^{\frac{5}{2}}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} - \frac{256 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} - \frac{1792 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}{- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}} - \frac{10 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 1024 x^{\frac{11}{2}} - 3072 x^{\frac{9}{2}} + 3072 x^{5} + 1024 x^{4}\right)} - \frac{x^{\frac{3}{2}}}{- 3072 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 9216 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 9216 x^{5} \sqrt{1 - \sqrt{x}} + 3072 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{32 x^{4}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{1600 x^{3} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} + \frac{896 x^{3} \sqrt{1 - \sqrt{x}}}{- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}} + \frac{64 x^{3}}{- 65536 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 2293760 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 458752 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 458752 x^{7} \sqrt{1 - \sqrt{x}} + 2293760 x^{6} \sqrt{1 - \sqrt{x}} + 1376256 x^{5} \sqrt{1 - \sqrt{x}} + 65536 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{1216 x^{2} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} + \frac{3456 x^{2} \sqrt{1 - \sqrt{x}}}{- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}} + \frac{32 x^{2}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{x^{2}}{- 3072 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 9216 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 9216 x^{5} \sqrt{1 - \sqrt{x}} + 3072 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{1024 x \sqrt{1 - \sqrt{x}}}{3 \left(- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}\right)} + \frac{8 x \sqrt{1 - \sqrt{x}}}{3 \left(- 1024 x^{\frac{11}{2}} - 3072 x^{\frac{9}{2}} + 3072 x^{5} + 1024 x^{4}\right)} + \frac{\sqrt{1 - \sqrt{x}}}{1536 x^{\frac{7}{2}} + 512 x^{\frac{5}{2}} - 512 x^{4} - 1536 x^{3}} - \frac{\sin^{2}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{128} - \frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos^{2}{\left(\frac{x}{2} \right)}}{64} - \frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{384} + \frac{\cos^{4}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{384} + \frac{\cos^{2}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{96} + \frac{1}{9216 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} + 3072 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} - 3072 x^{4} \sqrt{1 - \sqrt{x}} - 9216 x^{3} \sqrt{1 - \sqrt{x}}} + \frac{1}{4608 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} + 1536 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} - 1536 x^{4} \sqrt{1 - \sqrt{x}} - 4608 x^{3} \sqrt{1 - \sqrt{x}}} - \frac{1}{6 \left(- 256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 768 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 768 x^{3} \sqrt{1 - \sqrt{x}} + 256 x^{2} \sqrt{1 - \sqrt{x}}\right)} - \frac{1}{- 1024 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 3072 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 3072 x^{3} \sqrt{1 - \sqrt{x}} + 1024 x^{2} \sqrt{1 - \sqrt{x}}} + \frac{1}{- 512 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} + 512 x^{3} \sqrt{1 - \sqrt{x}}} - \frac{5}{256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{448 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} - \frac{128 x^{\frac{7}{2}}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} - \frac{704 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}{- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}} - \frac{8704 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}\right)} - \frac{128 x^{\frac{5}{2}}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} - \frac{256 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} - \frac{1792 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}{- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}} - \frac{10 x^{\frac{3}{2}} \sqrt{1 - \sqrt{x}}}{3 \left(- 1024 x^{\frac{11}{2}} - 3072 x^{\frac{9}{2}} + 3072 x^{5} + 1024 x^{4}\right)} - \frac{x^{\frac{3}{2}}}{- 3072 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 9216 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 9216 x^{5} \sqrt{1 - \sqrt{x}} + 3072 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{32 x^{4}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{1600 x^{3} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} + \frac{896 x^{3} \sqrt{1 - \sqrt{x}}}{- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}} + \frac{64 x^{3}}{- 65536 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 2293760 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 458752 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 458752 x^{7} \sqrt{1 - \sqrt{x}} + 2293760 x^{6} \sqrt{1 - \sqrt{x}} + 1376256 x^{5} \sqrt{1 - \sqrt{x}} + 65536 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{1216 x^{2} \sqrt{1 - \sqrt{x}}}{3 \left(- 65536 x^{\frac{15}{2}} - 1376256 x^{\frac{13}{2}} - 2293760 x^{\frac{11}{2}} - 458752 x^{\frac{9}{2}} + 458752 x^{7} + 2293760 x^{6} + 1376256 x^{5} + 65536 x^{4}\right)} + \frac{3456 x^{2} \sqrt{1 - \sqrt{x}}}{- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}} + \frac{32 x^{2}}{- 196608 x^{\frac{15}{2}} \sqrt{1 - \sqrt{x}} - 4128768 x^{\frac{13}{2}} \sqrt{1 - \sqrt{x}} - 6881280 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 1376256 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 1376256 x^{7} \sqrt{1 - \sqrt{x}} + 6881280 x^{6} \sqrt{1 - \sqrt{x}} + 4128768 x^{5} \sqrt{1 - \sqrt{x}} + 196608 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{x^{2}}{- 3072 x^{\frac{11}{2}} \sqrt{1 - \sqrt{x}} - 9216 x^{\frac{9}{2}} \sqrt{1 - \sqrt{x}} + 9216 x^{5} \sqrt{1 - \sqrt{x}} + 3072 x^{4} \sqrt{1 - \sqrt{x}}} + \frac{1024 x \sqrt{1 - \sqrt{x}}}{3 \left(- 393216 x^{\frac{13}{2}} - 1310720 x^{\frac{11}{2}} - 393216 x^{\frac{9}{2}} + 65536 x^{7} + 983040 x^{6} + 983040 x^{5} + 65536 x^{4}\right)} + \frac{8 x \sqrt{1 - \sqrt{x}}}{3 \left(- 1024 x^{\frac{11}{2}} - 3072 x^{\frac{9}{2}} + 3072 x^{5} + 1024 x^{4}\right)} + \frac{\sqrt{1 - \sqrt{x}}}{1536 x^{\frac{7}{2}} + 512 x^{\frac{5}{2}} - 512 x^{4} - 1536 x^{3}} - \frac{\sin^{2}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{128} - \frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\sin{\left(\frac{x}{2} \right)} \right)} \cos^{2}{\left(\frac{x}{2} \right)}}{64} - \frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{384} + \frac{\cos^{4}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{384} + \frac{\cos^{2}{\left(\frac{x}{2} \right)} \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{96} + \frac{1}{9216 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} + 3072 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} - 3072 x^{4} \sqrt{1 - \sqrt{x}} - 9216 x^{3} \sqrt{1 - \sqrt{x}}} + \frac{1}{4608 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} + 1536 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} - 1536 x^{4} \sqrt{1 - \sqrt{x}} - 4608 x^{3} \sqrt{1 - \sqrt{x}}} - \frac{1}{6 \left(- 256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 768 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 768 x^{3} \sqrt{1 - \sqrt{x}} + 256 x^{2} \sqrt{1 - \sqrt{x}}\right)} - \frac{1}{- 1024 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} - 3072 x^{\frac{5}{2}} \sqrt{1 - \sqrt{x}} + 3072 x^{3} \sqrt{1 - \sqrt{x}} + 1024 x^{2} \sqrt{1 - \sqrt{x}}} + \frac{1}{- 512 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}} + 512 x^{3} \sqrt{1 - \sqrt{x}}} - \frac{5}{256 x^{\frac{7}{2}} \sqrt{1 - \sqrt{x}}}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
/ ___________ \
| / ___ / /x\\|
|- \/ 1 - \/ x + cos|sin|-|||
| \ \2//|
lim |------------------------------|
x->0+| 4 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right)$$
oo
$$\infty$$
= 21599820.0325915
/ ___________ \
| / ___ / /x\\|
|- \/ 1 - \/ x + cos|sin|-|||
| \ \2//|
lim |------------------------------|
x->0-| 4 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right)$$
oo*I
$$\infty i$$
= (-432331.406565457 + 21136384.1773106j)
= (-432331.406565457 + 21136384.1773106j)
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right) = \infty$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right) = \cos{\left(\sin{\left(\frac{1}{2} \right)} \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right) = \cos{\left(\sin{\left(\frac{1}{2} \right)} \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right) = 0$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right) = \cos{\left(\sin{\left(\frac{1}{2} \right)} \right)}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right) = \cos{\left(\sin{\left(\frac{1}{2} \right)} \right)}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{- \sqrt{1 - \sqrt{x}} + \cos{\left(\sin{\left(\frac{x}{2} \right)} \right)}}{x^{4}}\right) = 0$$
Más detalles con x→-oo