Chapter 3 - Applications of Differentiation - 3.4 Limits at Infinity; Horizontal Asymptotes - 3.4 Exercises - Page 242: 16

$$1$$

Given $$ \lim _{x\to \infty } \frac{x^2}{\sqrt{x^4+1}} $$ Then \begin{aligned} \lim _{x\to \infty } \frac{x^2}{\sqrt{x^4+1}} &=\lim _{x\to \infty } \frac{\frac{x^2}{x^2}}{\sqrt{\frac{x^4}{x^4}+\frac{1}{x^4}}} \\ &= \lim _{x\to \infty } \frac{1}{\sqrt{1+\frac{1}{x^4}}} \\ &=\lim _{x\to \infty } \frac{1}{\sqrt{1+0}}\\ &=1 \end{aligned}