Find the limit of (ln(1+x)/(3(1+x)^(1/3)-1))^(x/(sin(x)^2)) as x approaches 0

 Exercise

$\lim_{x\to0}\left(\frac{\ln\left(1+x\right)}{3\left(1+x\right)^{\frac{1}{3}}-1}\right)^{\frac{x}{\sin^2\left(x\right)}}$

 

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

${\left(\lim_{x\to0}\left(\frac{\ln\left(1+x\right)}{3\sqrt[3]{1+x}-1}\right)\right)}^{\lim_{x\to0}\left(\frac{x}{\sin\left(x\right)^2}\right)}$

 

Evaluate the limit $\lim_{x\to0}\left(\frac{x}{\sin\left(x\right)^2}\right)$ by replacing all occurrences of $x$ by $0$

${\left(\lim_{x\to0}\left(\frac{\ln\left(1+x\right)}{3\sqrt[3]{1+x}-1}\right)\right)}^{\frac{0}{\sin\left(0\right)^2}}$

 

The sine of $0$ equals $0$

${\left(\lim_{x\to0}\left(\frac{\ln\left(1+x\right)}{3\sqrt[3]{1+x}-1}\right)\right)}^{\frac{0}{0^2}}$

 

Calculate the power $0^2$

$\lim_{x\to0}\left(\frac{\ln\left(1+x\right)}{3\sqrt[3]{1+x}-1}\right)^{\frac{0}{0}}$

 

$\frac{0}{0}$ represents an indeterminate form

indeterminate

indeterminate

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