Chapter 2 - Limits - 2.7 Limits at Infinity - Exercises - Page 83: 35

$$0$$

We evaluate the limit: \begin{aligned} & \lim_{x\to\infty}\sqrt{4 x^{4}+9 x}-2 x^{2} \\ &=\lim_{x\to\infty}\left(\sqrt{4 x^{4}+9 x}-2 x^{2}\right) \frac{\sqrt{4 x^{4}+9 x}+2 x^{2}}{\sqrt{4 x^{4}+9 x}+2 x^{2}} \\ &=\lim_{x\to\infty}\frac{\left(4 x^{4}+9 x\right)-4 x^{4}}{\sqrt{4 x^{4}+9 x}+2 x^{2}}\\ &=\lim_{x\to\infty}\frac{9 x}{\sqrt{4 x^{4}+9 x}+2 x^{2}} \\ &=0\end{aligned}