Find the limit of (1/(e^(6x)-1))^(7/x) as x approaches infinity

 Exercise

$\lim_{x\to\infty}\left(\left(\frac{1}{e^{6x}-1}\right)^{\frac{7}{x}}\right)$

 

Apply the power rule of limits: $\displaystyle{\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\displaystyle\lim_{x\to a}g(x)}}$

${\left(\lim_{x\to\infty }\left(\frac{1}{e^{6x}-1}\right)\right)}^{\lim_{x\to\infty }\left(\frac{7}{x}\right)}$

 

Evaluate the limit $\lim_{x\to\infty }\left(\frac{7}{x}\right)$ by replacing all occurrences of $x$ by $\infty $

${\left(\lim_{x\to\infty }\left(\frac{1}{e^{6x}-1}\right)\right)}^0$

 

Evaluate the limit $\lim_{x\to\infty }\left(\frac{1}{e^{6x}-1}\right)$ by replacing all occurrences of $x$ by $\infty $

$0^0$

 

$0^0$ represents an indeterminate form

indeterminate

indeterminate

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