lhopital: problem 3
(1 point)
evaluate the limit
lim _ { x
ightarrow infty } sqrt { x ^ { 2 } + 7 x + 2 } - x
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Question

lhopital: problem 3
(1 point)
evaluate the limit
 lim _ { x
ightarrow infty } sqrt { x ^ { 2 } + 7 x + 2 } - x
preview my answers submit answers
you have attempted this problem 0 times.
you have unlimited attempts remaining.
email instructor
page generated october 20, 2025 at 12:01:00 pm cdt
webwork © 1996-2024 | theme: math4 | ww_version: 2.19 | pg_version: 2.19
the webwork project

Accepted Answer

# Explanation:
## Step1: Rationalize the expression
Multiply and divide by the conjugate $\sqrt{x^{2}+7x + 2}+x$
$$
\begin{align*}
\lim_{x
ightarrow\infty}(\sqrt{x^{2}+7x + 2}-x)&=\lim_{x
ightarrow\infty}\frac{(\sqrt{x^{2}+7x + 2}-x)(\sqrt{x^{2}+7x + 2}+x)}{\sqrt{x^{2}+7x + 2}+x}\
&=\lim_{x
ightarrow\infty}\frac{(x^{2}+7x + 2)-x^{2}}{\sqrt{x^{2}+7x + 2}+x}\
&=\lim_{x
ightarrow\infty}\frac{7x+2}{\sqrt{x^{2}+7x + 2}+x}
\end{align*}
$$
## Step2: Divide numerator and denominator by $x$
$$
\begin{align*}
\lim_{x
ightarrow\infty}\frac{7x+2}{\sqrt{x^{2}+7x + 2}+x}&=\lim_{x
ightarrow\infty}\frac{7+\frac{2}{x}}{\sqrt{1+\frac{7}{x}+\frac{2}{x^{2}}}+1}\
\end{align*}
$$
## Step3: Evaluate the limit
As $x
ightarrow\infty$, $\frac{2}{x}
ightarrow0$, $\frac{7}{x}
ightarrow0$ and $\frac{2}{x^{2}}
ightarrow0$
$$
\begin{align*}
\lim_{x
ightarrow\infty}\frac{7+\frac{2}{x}}{\sqrt{1+\frac{7}{x}+\frac{2}{x^{2}}}+1}&=\frac{7 + 0}{\sqrt{1+0+0}+1}\
&=\frac{7}{2}
\end{align*}
$$
# Answer:
$\frac{7}{2}$

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Question

Explanation:

Step1: Rationalize the expression

Step2: Divide numerator and denominator by \(x\)

Step3: Evaluate the limit

Answer: