## Limit at Infinity Problems with Square Roots

### Limits at Infinity with Square Roots: Problems and Solutions

To analyze limit at infinity problems with square roots, we’ll use the tools we used earlier to solve limit at infinity problems, PLUS one additional bit: it is *crucial* to remember

\[ \bbox[yellow,5px]{

\begin{align*}

\text{If $x$ is positive: } x &= \sqrt{x^2} \\[8px]
\text{If $x$ is negative: } x &= -\sqrt{x^2} \\[8px]
\end{align*} } \]

• By contrast, if $x = -3$, then $x = -3 = -\sqrt{9}$.

You *must* remember that $x = -\sqrt{x^2}$ in any problem where $x \to\, -\infty$, since you’re then automatically looking at negative values of *x*.

The problems below illustrate, starting with part (b) of the first one.

**(a)**Find $\displaystyle{\lim_{x \to \infty}\frac{\sqrt{5x^2 + 2x}}{x}}.$

**(b)** Find $\displaystyle{\lim_{x \to\, -\infty}\frac{\sqrt{5x^2 + 2x}}{x}}.$

*We think this problem has a cool, surprising result.*

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

*This problem is by student request. It has another (the same, actually) cool, surprising result.*

Find $\displaystyle{\lim_{x \to \infty}\left(\sqrt{x + \sqrt{x}} – \sqrt{x} \right)}$.

*This is a generalized version of Problem #2 above.*

Find $\displaystyle{\lim_{x \to \infty}\left(\sqrt{a^2 x^2 + x} -ax \right)},$ where $a$ is a positive constant.

*A student posted this question in the Comments below.*

Find $\displaystyle{\lim_{x \to \infty}\big[\sqrt{x}\sqrt{x}\sqrt{x} – \sqrt{x} \big]}.$

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√x√x√x-√x Lim app to infinity is

We’ve updated the page to provide the solution to this question as Practice Problem #7 above. It’s a good problem, and we hope we’ve been able to help you (and future students) solve it successfully for yourself.

Thanks for asking!