Answer
$$2$$
Work Step by Step
Given $$ \lim _{x\to -\infty }\frac{\sqrt{1+4x^6}}{2-x^3}$$
Then
\begin{aligned}
\lim _{x\to -\infty }\frac{\sqrt{1+4x^6}}{2-x^3}&=\lim _{x\to -\infty }\frac{\sqrt{\frac{1}{x^6}+\frac{4x^6}{x^6}}}{\frac{2}{x^3}-\frac{x^3}{x^3}} \\
&=\lim _{x\to -\infty }\frac{\sqrt{\frac{1}{x^6}+4}}{\frac{2}{x^3}-1}\\
&=\lim _{x\to- \infty }\frac{\sqrt{0+4}}{0-1}\\
&= 2
\end{aligned}
