Chapter 1 - Limits and Continuity - 1.3 Limits At Infinity; End Behavior Of A Function - Exercises Set 1.3 - Page 79: 30

$$ \sqrt{3}$$

Given $$\lim _{x \rightarrow \infty}\frac{\sqrt { 3x^4+x}}{\sqrt {x^2-8} } $$ Then \begin{align*} \lim _{x \rightarrow \infty}\frac{\sqrt { 3x^4+x}}{\sqrt {x^2-8} } &=\lim _{x \rightarrow \infty}\frac{\sqrt { 3x^4/x^4+x/x^4}}{\sqrt {x^2/x^2-8/x^2} } \\ &=\lim _{x \rightarrow \infty}\frac{\sqrt { 3 +1/x^3}}{\sqrt {1-8/x^2} } \\ &= \sqrt{3} \end{align*}