evaluate the limit\n$$ lim _ { x \rightarrow infty } \frac { sqrt { 10 + 8 x ^ { 2 } } } { ( 5 + 11 x ) } $$
evaluate the limit\n$$ lim _ { x \rightarrow infty } \frac { sqrt { 10 + 8 x ^ { 2 } } } { ( 5 + 11 x ) } $$
Answer
Explanation:
Step1: Divide numerator and denominator by (x)
$$ \begin{align*} \lim_{x\rightarrow\infty}\frac{\sqrt{10 + 8x^{2}}}{5+11x}&=\lim_{x\rightarrow\infty}\frac{\sqrt{\frac{10}{x^{2}}+ 8}}{\frac{5}{x}+11}\ \end{align*} $$
Step2: Apply the limit
As (x\rightarrow\infty), (\lim_{x\rightarrow\infty}\frac{10}{x^{2}} = 0) and (\lim_{x\rightarrow\infty}\frac{5}{x}=0) $$ \begin{align*} \lim_{x\rightarrow\infty}\frac{\sqrt{\frac{10}{x^{2}}+ 8}}{\frac{5}{x}+11}&=\frac{\sqrt{0 + 8}}{0+11}\ &=\frac{\sqrt{8}}{11}\ &=\frac{2\sqrt{2}}{11} \end{align*} $$
Answer:
(\frac{2\sqrt{2}}{11})