## Limit at Infinity Problems and Solutions

### Limits at Infinity: Problems and Solutions

Are you working to solve problems about $\displaystyle{\lim_{x \to \infty}}$ and $\displaystyle{\lim_{x \to\, -\infty}}$? Let’s look at common limit at infinity problems and solutions so you can learn to solve them routinely.

**(a)**Find $\displaystyle{\lim_{x \to \infty} \left(3x^3 + 947x^2 – \sqrt{x} \right)}.$

**(b)**Find $\displaystyle{\lim_{x \to -\infty} \left(3x^3 + 947x^2 – \sqrt{x} \right)}.$

**(a)**Find $\displaystyle{\lim_{x \to \infty} \sin(x)}$.

**(b)**Find $\displaystyle{\lim_{x \to -\infty} \cos(x)}$.

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## What are your thoughts and questions?

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

Please see the reply below.

Question is : lim x→0 (1/x)^(1/x)

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!