## Limit at Infinity Problems and Solutions

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Vikas Bharti
1 year ago

Or We can say : lim x→0 (1+(1/x))^(1/x) = ?

Matheno
Editor
Reply to  Vikas Bharti
1 year ago

Vikas Bharti
1 year ago

Question is : lim x→0 (1/x)^(1/x)

Matheno
Editor
Reply to  Vikas Bharti
1 year ago

This is a classic L’Hôpital’s Rule question, which this page doesn’t address. But in case it helps, here are the steps you can use: First let $y = (1/x)^{(1/x)}$. Then
\begin{align*}
\ln y &= \ln(1/x)^{(1/x)} \8px] &= (1/x) \ln (1/x) \\[8px] &= -\frac{\ln (x)}{x} \end{align*} where in the second line we made use of the fact that \ln a^b = b \ln a, and in the third line \ln(1/x) = -\ln x. From there, if you take  \displaystyle{\lim_{x \to \infty }} on both sides, on the right you have \dfrac{\infty}{\infty} and so you can apply L’Hôpital’s Rule. You should find \[\displaystyle{\lim_{x \to \infty } \ln y = 0}
and so when you undo the natural log, you find that the initial limit = 1.

We hope that helps!