Question

evaluate the following limits. if needed, enter inf for ∞ and minf for -∞.
(a)
\\(\lim_{x\to\infty}(\sqrt{x^{2}+5x + 1}-x)=\\)
(b)
\\(\lim_{x\to -\infty}(\sqrt{x^{2}+5x + 1}-x)=\\)

Explanation:

Step1: Rationalize the expression

Multiply and divide by $\sqrt{x^{2}+5x + 1}+x$.

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Step2: Divide numerator and denominator by $x$

As $x
ightarrow\infty$, $x>0$, so $\sqrt{x^{2}}=x$.

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Step3: Evaluate the limit

Using the limit rules $\lim_{x
ightarrow\infty}\frac{1}{x}=0$ and $\lim_{x
ightarrow\infty}\frac{1}{x^{2}} = 0$.

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Step4: For $x

ightarrow-\infty$
Multiply and divide by $\sqrt{x^{2}+5x + 1}+x$.

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Since $x
ightarrow-\infty$, $\sqrt{x^{2}}=-x$. Divide numerator and denominator by $-x$.

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Answer:

(a) $\frac{5}{2}$
(b) INF