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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

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

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



Tips iconYou 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.

Practice Problem #1
(a) Find $\displaystyle{\lim_{x \to \infty}\frac{\sqrt{5x^2 + 2x}}{x}}.$

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

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Practice Problem #2
We think this problem has a cool, surprising result.
Find $\displaystyle{\lim_{x \to \infty}\left(\sqrt{x^2 + x} – x \right)}.$

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Practice Problem #3
This problem is by student request. It has another (the same, actually) cool, surprising result.
Find $\displaystyle{\lim_{x \to \infty}\left(\sqrt{x + \sqrt{x}} – \sqrt{x} \right)}$.
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Practice Problem #4
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.
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Practice Problem #5
Find $\displaystyle{\lim_{x \to \infty} \left(\sqrt{x^2 + ax} – \sqrt{x^2 + bx} \right)},$ where $a$ and $b$ are constants.

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Practice Problem #6
Find $\displaystyle{\lim_{x \to \, -\infty}\left(x + \sqrt{x^2 + ax} \right) }$, where $a$ is a constant.
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