Use L'Hobital's rule to evaluate the given limit if the limit is an indeterminate form.

$\lim _{x \rightarrow 0}(1+2 x)^{1 / x}$

Question

Use L'Hobital's rule to evaluate the given limit if the limit is an indeterminate form.

$\lim _{x \rightarrow 0}(1+2 x)^{1 / x}$

Quizlet Admin · Accepted Answer

To get the specified limit. We can be rewritten the given limit as:
$$\begin{aligned}
\lim_{xightarrow0}e^{\frac{1}{x}\ln\lparen1+2xparen}&=e^{\lim_{xightarrow0}\frac{\ln\lparen1+2xparen}{x}}\
&=e^{\frac{0}{0}}\end{aligned}$$
So the limit in the exponent; $\lim_{xightarrow0}\frac{\ln\lparen1+2xparen}{x}$ is in the indeterminate form $\frac{0}{0}$.

Use L'Hobital's rule to evaluate the given limit if the limit is an indeterminate form.

$\lim _{x \rightarrow 0}(1+2 x)^{1 / x}$

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Use L'Hobital's rule to evaluate the given limit if the limit is an indeterminate form. lim⁡x→0(1+2x)1/x\lim _{x \rightarrow 0}(1+2 x)^{1 / x}limx→0​(1+2x)1/x

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Use L'Hobital's rule to evaluate the given limit if the limit is an indeterminate form.

$\lim _{x \rightarrow 0}(1+2 x)^{1 / x}$