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Expresiones idénticas
- (tres ^n+ cinco ^n)/(tres ^(uno +n)- cinco ^n)
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- (tres en el grado n más cinco en el grado n) dividir por (tres en el grado (uno más n) menos cinco en el grado n)
- (3n+5n)/(3(1+n)-5n)
- 3n+5n/31+n-5n
- 3^n+5^n/3^1+n-5^n
- (3^n+5^n) dividir por (3^(1+n)-5^n)
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Expresiones semejantes
- (3^n+5^n)/(3^(1+n)+5^n)
- (3^n+5^n)/(3^(1-n)-5^n)
- (3^n-5^n)/(3^(1+n)-5^n)
Límite de la función (3^n+5^n)/(3^(1+n)-5^n)
Solución
Ha introducido
[src]
/ n n \
| 3 + 5 |
lim |-----------|
n->-oo| 1 + n n|
\3 - 5 /
$$\lim_{n \to -\infty}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right)$$
Limit((3^n + 5^n)/(3^(1 + n) - 5^n), n, -oo)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{n \to -\infty} \frac{1}{3 \cdot 3^{n} - 5^{n}} = \infty$$
y el límite para el denominador es
$$\lim_{n \to -\infty} \frac{1}{3^{n} + 5^{n}} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to -\infty}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right)$$
=
$$\lim_{n \to -\infty}\left(\frac{\frac{d}{d n} \frac{1}{3 \cdot 3^{n} - 5^{n}}}{\frac{d}{d n} \frac{1}{3^{n} + 5^{n}}}\right)$$
=
$$\lim_{n \to -\infty}\left(\frac{1}{\left(- 3^{n} \log{\left(3 \right)} - 5^{n} \log{\left(5 \right)}\right) \left(- \frac{6 \cdot 15^{n}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{9 \cdot 3^{2 n}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{5^{2 n}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}}\right)}\right)$$
=
$$\lim_{n \to -\infty}\left(\frac{\frac{d}{d n} \frac{1}{- 3^{n} \log{\left(3 \right)} - 5^{n} \log{\left(5 \right)}}}{\frac{d}{d n} \left(- \frac{6 \cdot 15^{n}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{9 \cdot 3^{2 n}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{5^{2 n}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}}\right)}\right)$$
=
$$\lim_{n \to -\infty}\left(\frac{3^{n} \log{\left(3 \right)}^{2} + 5^{n} \log{\left(5 \right)}^{2}}{\left(- 3^{n} \log{\left(3 \right)} - 5^{n} \log{\left(5 \right)}\right)^{2} \left(- \frac{6 \cdot 15^{n} \log{\left(15 \right)}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} - \frac{6 \cdot 15^{n} \left(9 \cdot 3^{3 n} \log{\left(3 \right)}^{2} - 2 \cdot 3^{2 n} 5^{n} \log{\left(3 \right)} \log{\left(5 \right)} - 3^{2 n} 5^{n} \log{\left(5 \right)}^{2} + 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)}^{2} + 6 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} \log{\left(5 \right)} + 6 \cdot 45^{n} \log{\left(3 \right)} \log{\left(45 \right)} - 3 \cdot 5^{3 n} \log{\left(5 \right)}^{2} - 2 \cdot 75^{n} \log{\left(5 \right)} \log{\left(75 \right)}\right)}{\left(- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}\right)^{2}} + \frac{18 \cdot 3^{2 n} \log{\left(3 \right)}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{9 \cdot 3^{2 n} \left(9 \cdot 3^{3 n} \log{\left(3 \right)}^{2} - 2 \cdot 3^{2 n} 5^{n} \log{\left(3 \right)} \log{\left(5 \right)} - 3^{2 n} 5^{n} \log{\left(5 \right)}^{2} + 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)}^{2} + 6 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} \log{\left(5 \right)} + 6 \cdot 45^{n} \log{\left(3 \right)} \log{\left(45 \right)} - 3 \cdot 5^{3 n} \log{\left(5 \right)}^{2} - 2 \cdot 75^{n} \log{\left(5 \right)} \log{\left(75 \right)}\right)}{\left(- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}\right)^{2}} + \frac{2 \cdot 5^{2 n} \log{\left(5 \right)}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{5^{2 n} \left(9 \cdot 3^{3 n} \log{\left(3 \right)}^{2} - 2 \cdot 3^{2 n} 5^{n} \log{\left(3 \right)} \log{\left(5 \right)} - 3^{2 n} 5^{n} \log{\left(5 \right)}^{2} + 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)}^{2} + 6 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} \log{\left(5 \right)} + 6 \cdot 45^{n} \log{\left(3 \right)} \log{\left(45 \right)} - 3 \cdot 5^{3 n} \log{\left(5 \right)}^{2} - 2 \cdot 75^{n} \log{\left(5 \right)} \log{\left(75 \right)}\right)}{\left(- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}\right)^{2}}\right)}\right)$$
=
$$\lim_{n \to -\infty}\left(\frac{3^{n} \log{\left(3 \right)}^{2} + 5^{n} \log{\left(5 \right)}^{2}}{\left(- 3^{n} \log{\left(3 \right)} - 5^{n} \log{\left(5 \right)}\right)^{2} \left(- \frac{6 \cdot 15^{n} \log{\left(15 \right)}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} - \frac{6 \cdot 15^{n} \left(9 \cdot 3^{3 n} \log{\left(3 \right)}^{2} - 2 \cdot 3^{2 n} 5^{n} \log{\left(3 \right)} \log{\left(5 \right)} - 3^{2 n} 5^{n} \log{\left(5 \right)}^{2} + 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)}^{2} + 6 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} \log{\left(5 \right)} + 6 \cdot 45^{n} \log{\left(3 \right)} \log{\left(45 \right)} - 3 \cdot 5^{3 n} \log{\left(5 \right)}^{2} - 2 \cdot 75^{n} \log{\left(5 \right)} \log{\left(75 \right)}\right)}{\left(- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}\right)^{2}} + \frac{18 \cdot 3^{2 n} \log{\left(3 \right)}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{9 \cdot 3^{2 n} \left(9 \cdot 3^{3 n} \log{\left(3 \right)}^{2} - 2 \cdot 3^{2 n} 5^{n} \log{\left(3 \right)} \log{\left(5 \right)} - 3^{2 n} 5^{n} \log{\left(5 \right)}^{2} + 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)}^{2} + 6 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} \log{\left(5 \right)} + 6 \cdot 45^{n} \log{\left(3 \right)} \log{\left(45 \right)} - 3 \cdot 5^{3 n} \log{\left(5 \right)}^{2} - 2 \cdot 75^{n} \log{\left(5 \right)} \log{\left(75 \right)}\right)}{\left(- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}\right)^{2}} + \frac{2 \cdot 5^{2 n} \log{\left(5 \right)}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{5^{2 n} \left(9 \cdot 3^{3 n} \log{\left(3 \right)}^{2} - 2 \cdot 3^{2 n} 5^{n} \log{\left(3 \right)} \log{\left(5 \right)} - 3^{2 n} 5^{n} \log{\left(5 \right)}^{2} + 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)}^{2} + 6 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} \log{\left(5 \right)} + 6 \cdot 45^{n} \log{\left(3 \right)} \log{\left(45 \right)} - 3 \cdot 5^{3 n} \log{\left(5 \right)}^{2} - 2 \cdot 75^{n} \log{\left(5 \right)} \log{\left(75 \right)}\right)}{\left(- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}\right)^{2}}\right)}\right)$$
=
$$\frac{1}{3}$$
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
$$\lim_{n \to -\infty} \frac{1}{3 \cdot 3^{n} - 5^{n}} = \infty$$
y el límite para el denominador es
$$\lim_{n \to -\infty} \frac{1}{3^{n} + 5^{n}} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to -\infty}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right)$$
=
$$\lim_{n \to -\infty}\left(\frac{\frac{d}{d n} \frac{1}{3 \cdot 3^{n} - 5^{n}}}{\frac{d}{d n} \frac{1}{3^{n} + 5^{n}}}\right)$$
=
$$\lim_{n \to -\infty}\left(\frac{1}{\left(- 3^{n} \log{\left(3 \right)} - 5^{n} \log{\left(5 \right)}\right) \left(- \frac{6 \cdot 15^{n}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{9 \cdot 3^{2 n}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{5^{2 n}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}}\right)}\right)$$
=
$$\lim_{n \to -\infty}\left(\frac{\frac{d}{d n} \frac{1}{- 3^{n} \log{\left(3 \right)} - 5^{n} \log{\left(5 \right)}}}{\frac{d}{d n} \left(- \frac{6 \cdot 15^{n}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{9 \cdot 3^{2 n}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{5^{2 n}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}}\right)}\right)$$
=
$$\lim_{n \to -\infty}\left(\frac{3^{n} \log{\left(3 \right)}^{2} + 5^{n} \log{\left(5 \right)}^{2}}{\left(- 3^{n} \log{\left(3 \right)} - 5^{n} \log{\left(5 \right)}\right)^{2} \left(- \frac{6 \cdot 15^{n} \log{\left(15 \right)}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} - \frac{6 \cdot 15^{n} \left(9 \cdot 3^{3 n} \log{\left(3 \right)}^{2} - 2 \cdot 3^{2 n} 5^{n} \log{\left(3 \right)} \log{\left(5 \right)} - 3^{2 n} 5^{n} \log{\left(5 \right)}^{2} + 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)}^{2} + 6 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} \log{\left(5 \right)} + 6 \cdot 45^{n} \log{\left(3 \right)} \log{\left(45 \right)} - 3 \cdot 5^{3 n} \log{\left(5 \right)}^{2} - 2 \cdot 75^{n} \log{\left(5 \right)} \log{\left(75 \right)}\right)}{\left(- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}\right)^{2}} + \frac{18 \cdot 3^{2 n} \log{\left(3 \right)}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{9 \cdot 3^{2 n} \left(9 \cdot 3^{3 n} \log{\left(3 \right)}^{2} - 2 \cdot 3^{2 n} 5^{n} \log{\left(3 \right)} \log{\left(5 \right)} - 3^{2 n} 5^{n} \log{\left(5 \right)}^{2} + 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)}^{2} + 6 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} \log{\left(5 \right)} + 6 \cdot 45^{n} \log{\left(3 \right)} \log{\left(45 \right)} - 3 \cdot 5^{3 n} \log{\left(5 \right)}^{2} - 2 \cdot 75^{n} \log{\left(5 \right)} \log{\left(75 \right)}\right)}{\left(- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}\right)^{2}} + \frac{2 \cdot 5^{2 n} \log{\left(5 \right)}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{5^{2 n} \left(9 \cdot 3^{3 n} \log{\left(3 \right)}^{2} - 2 \cdot 3^{2 n} 5^{n} \log{\left(3 \right)} \log{\left(5 \right)} - 3^{2 n} 5^{n} \log{\left(5 \right)}^{2} + 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)}^{2} + 6 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} \log{\left(5 \right)} + 6 \cdot 45^{n} \log{\left(3 \right)} \log{\left(45 \right)} - 3 \cdot 5^{3 n} \log{\left(5 \right)}^{2} - 2 \cdot 75^{n} \log{\left(5 \right)} \log{\left(75 \right)}\right)}{\left(- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}\right)^{2}}\right)}\right)$$
=
$$\lim_{n \to -\infty}\left(\frac{3^{n} \log{\left(3 \right)}^{2} + 5^{n} \log{\left(5 \right)}^{2}}{\left(- 3^{n} \log{\left(3 \right)} - 5^{n} \log{\left(5 \right)}\right)^{2} \left(- \frac{6 \cdot 15^{n} \log{\left(15 \right)}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} - \frac{6 \cdot 15^{n} \left(9 \cdot 3^{3 n} \log{\left(3 \right)}^{2} - 2 \cdot 3^{2 n} 5^{n} \log{\left(3 \right)} \log{\left(5 \right)} - 3^{2 n} 5^{n} \log{\left(5 \right)}^{2} + 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)}^{2} + 6 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} \log{\left(5 \right)} + 6 \cdot 45^{n} \log{\left(3 \right)} \log{\left(45 \right)} - 3 \cdot 5^{3 n} \log{\left(5 \right)}^{2} - 2 \cdot 75^{n} \log{\left(5 \right)} \log{\left(75 \right)}\right)}{\left(- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}\right)^{2}} + \frac{18 \cdot 3^{2 n} \log{\left(3 \right)}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{9 \cdot 3^{2 n} \left(9 \cdot 3^{3 n} \log{\left(3 \right)}^{2} - 2 \cdot 3^{2 n} 5^{n} \log{\left(3 \right)} \log{\left(5 \right)} - 3^{2 n} 5^{n} \log{\left(5 \right)}^{2} + 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)}^{2} + 6 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} \log{\left(5 \right)} + 6 \cdot 45^{n} \log{\left(3 \right)} \log{\left(45 \right)} - 3 \cdot 5^{3 n} \log{\left(5 \right)}^{2} - 2 \cdot 75^{n} \log{\left(5 \right)} \log{\left(75 \right)}\right)}{\left(- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}\right)^{2}} + \frac{2 \cdot 5^{2 n} \log{\left(5 \right)}}{- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}} + \frac{5^{2 n} \left(9 \cdot 3^{3 n} \log{\left(3 \right)}^{2} - 2 \cdot 3^{2 n} 5^{n} \log{\left(3 \right)} \log{\left(5 \right)} - 3^{2 n} 5^{n} \log{\left(5 \right)}^{2} + 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)}^{2} + 6 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} \log{\left(5 \right)} + 6 \cdot 45^{n} \log{\left(3 \right)} \log{\left(45 \right)} - 3 \cdot 5^{3 n} \log{\left(5 \right)}^{2} - 2 \cdot 75^{n} \log{\left(5 \right)} \log{\left(75 \right)}\right)}{\left(- 3 \cdot 3^{3 n} \log{\left(3 \right)} + 3^{2 n} 5^{n} \log{\left(5 \right)} - 3 \cdot 3^{n} 5^{2 n} \log{\left(3 \right)} - 6 \cdot 45^{n} \log{\left(3 \right)} + 5^{3 n} \log{\left(5 \right)} + 2 \cdot 75^{n} \log{\left(5 \right)}\right)^{2}}\right)}\right)$$
=
$$\frac{1}{3}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
Gráfica
Otros límites con n→0, -oo, +oo, 1
$$\lim_{n \to -\infty}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right) = \frac{1}{3}$$
$$\lim_{n \to \infty}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right) = -1$$
Más detalles con n→oo
$$\lim_{n \to 0^-}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right) = 1$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right) = 1$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right) = 2$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right) = 2$$
Más detalles con n→1 a la derecha
$$\lim_{n \to \infty}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right) = -1$$
Más detalles con n→oo
$$\lim_{n \to 0^-}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right) = 1$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right) = 1$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right) = 2$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{3^{n} + 5^{n}}{3^{n + 1} - 5^{n}}\right) = 2$$
Más detalles con n→1 a la derecha