evaluate the limit\nlim (sqrt(x^2 + 2) - sqrt(x^2 - 5))\nx->inf\nanswer:\nsubmit answer next item

Explanation:

Step1: Rationalize the expression

Multiply and divide by $\sqrt{x^{2}+2}+\sqrt{x^{2}-5}$: [ \begin{align*} &\lim_{x\rightarrow\infty}(\sqrt{x^{2}+2}-\sqrt{x^{2}-5})\times\frac{\sqrt{x^{2}+2}+\sqrt{x^{2}-5}}{\sqrt{x^{2}+2}+\sqrt{x^{2}-5}}\ =&\lim_{x\rightarrow\infty}\frac{(x^{2}+2)-(x^{2}-5)}{\sqrt{x^{2}+2}+\sqrt{x^{2}-5}}\ =&\lim_{x\rightarrow\infty}\frac{7}{\sqrt{x^{2}+2}+\sqrt{x^{2}-5}} \end{align*} ]

Step2: Divide numerator and denominator by $x$

Since $x\rightarrow\infty$, we have: [ \begin{align*} &\lim_{x\rightarrow\infty}\frac{7/x}{\sqrt{1 + \frac{2}{x^{2}}}+\sqrt{1-\frac{5}{x^{2}}}} \end{align*} ] As $x\rightarrow\infty$, $\frac{2}{x^{2}}\rightarrow0$ and $\frac{5}{x^{2}}\rightarrow0$.

Step3: Evaluate the limit

[ \begin{align*} &\frac{0}{\sqrt{1 + 0}+\sqrt{1-0}}=0 \end{align*} ]

Answer:

$0$