Question

$lim_{x
ightarrowinfty}\frac{sqrt{6 + 5x^{2}}}{(8 + 11x)}$

Explanation:

Step1: Divide numerator and denominator by \(x\)

As \(x\to\infty\), we know that for \(x>0\), \(\sqrt{6 + 5x^{2}}=x\sqrt{\frac{6}{x^{2}}+5}\). So, \(\lim_{x\to\infty}\frac{\sqrt{6 + 5x^{2}}}{8 + 11x}=\lim_{x\to\infty}\frac{x\sqrt{\frac{6}{x^{2}}+5}}{x(\frac{8}{x}+11)}\)

Step2: Simplify the expression

Cancel out the common - factor \(x\) in the numerator and denominator: \(\lim_{x\to\infty}\frac{\sqrt{\frac{6}{x^{2}}+5}}{\frac{8}{x}+11}\)

Step3: Use the limit property \(\lim_{x\to\infty}\frac{1}{x}=0\)

We know that \(\lim_{x\to\infty}\frac{6}{x^{2}} = 6\lim_{x\to\infty}(\frac{1}{x})^{2}=0\) and \(\lim_{x\to\infty}\frac{8}{x}=0\). Then \(\lim_{x\to\infty}\frac{\sqrt{\frac{6}{x^{2}}+5}}{\frac{8}{x}+11}=\frac{\sqrt{0 + 5}}{0+11}\)

Answer:

\(\frac{\sqrt{5}}{11}\)