**Update:** We now have *much* more interactive ways for you to learn about the foundational concept of Limits, making heavy use of Desmos graphing calculators so you can work with these ideas for yourself, and develop your problem solving skills step-by-step. Please visit our Limits Chapter to *really* get this material down for yourself.

It’s all free, and waiting for you! (Why? Just because we’re educators who believe you deserve the chance to develop a better understanding of Calculus for yourself, and so we’re aiming to provide that. We hope you’ll take advantage!)

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.

We'd love to hear:

- What questions do you have about the solutions above?
- Which ones are giving you the most trouble?
- What other Limits problems are you trying to work through for your class?

As of September 2022, we’re using our Forum for comments and discussion of this topic, and for any math questions. We’d love to see you there! Please tap to visit our Forum: community.matheno.com.

☕

Buy us a coffeeIf we've helped, please considergiving a little something back.

Thank you! 😊