evaluate the limit\nlim (sqrt(x^2 + 5) - sqrt(x^2 - 6))\nx->inf\nanswer:\nvideo example: solving a similar…

Explanation:

Step1: Rationalize the expression

Multiply and divide by $\sqrt{x^{2}+5}+\sqrt{x^{2}-6}$: [ \begin{align*} &\lim_{x\rightarrow\infty}(\sqrt{x^{2}+5}-\sqrt{x^{2}-6})\times\frac{\sqrt{x^{2}+5}+\sqrt{x^{2}-6}}{\sqrt{x^{2}+5}+\sqrt{x^{2}-6}}\ =&\lim_{x\rightarrow\infty}\frac{(x^{2}+5)-(x^{2}-6)}{\sqrt{x^{2}+5}+\sqrt{x^{2}-6}}\ =&\lim_{x\rightarrow\infty}\frac{x^{2}+5 - x^{2}+6}{\sqrt{x^{2}+5}+\sqrt{x^{2}-6}}\ =&\lim_{x\rightarrow\infty}\frac{11}{\sqrt{x^{2}+5}+\sqrt{x^{2}-6}} \end{align*} ]

Step2: Divide numerator and denominator by $x$

Since $x\rightarrow\infty$, we know that $x>0$. So $\sqrt{x^{2}} = x$. [ \begin{align*} &\lim_{x\rightarrow\infty}\frac{11/x}{\sqrt{1 + 5/x^{2}}+\sqrt{1-6/x^{2}}}\ \end{align*} ] As $x\rightarrow\infty$, $\frac{5}{x^{2}}\rightarrow0$ and $\frac{6}{x^{2}}\rightarrow0$.

Step3: Evaluate the limit

[ \begin{align*} &\frac{\lim_{x\rightarrow\infty}(11/x)}{\lim_{x\rightarrow\infty}\sqrt{1 + 5/x^{2}}+\lim_{x\rightarrow\infty}\sqrt{1-6/x^{2}}}\ =&\frac{0}{\sqrt{1 + 0}+\sqrt{1-0}}\ =&0 \end{align*} ]

Answer:

$0$