If you just need practice with limits problems for now, previous students have found what’s below super-helpful. And if you have questions, please ask on our Forum!

Summary

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 \[ \bbox[yellow,5px]{
\begin{align*}
\text{If $x$ is positive: } x &= \sqrt{x^2} \\[8px]
\text{If $x$ is negative: } x &= -\sqrt{x^2} \\[8px]
\end{align*} } \]

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.

• For example, if $x = 3$, then $x = 3 = \sqrt{9}$.

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

You

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

For a fuller discussion of this crucial point, please visit the screen “Limit at Infinity with Square Roots” in our Limits Chapter devoted to this topic. We also have specifically-designed interactive Desmos graphing calculators there that will help you understand what it is you’re doing when you compute these limits.

Problem #1

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

Problem #2

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

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

Problem #3

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

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

Problem #4

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

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

Problem #5

Find $\displaystyle{\lim_{x \to \infty} \left(\sqrt{x^2 + ax} - \sqrt{x^2 + bx} \right)},$ where $a$ and $b$ are constants.

Problem #6

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

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