Question

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

limx0(1+2x)1/x\lim _{x \rightarrow 0}(1+2 x)^{1 / x}

Solution

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Answered 1 year ago
Answered 1 year ago
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To get the specified limit. We can be rewritten the given limit as:

limx0e1xln(1+2x)=elimx0ln(1+2x)x=e00\begin{aligned} \lim_{x\rightarrow0}e^{\frac{1}{x}\ln\lparen1+2x\rparen}&=e^{\lim_{x\rightarrow0}\frac{\ln\lparen1+2x\rparen}{x}}\\ &=e^{\frac{0}{0}}\end{aligned}

So the limit in the exponent; limx0ln(1+2x)x\lim_{x\rightarrow0}\frac{\ln\lparen1+2x\rparen}{x} is in the indeterminate form 00\frac{0}{0}.

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