Chapter 1 - Limits and Continuity - 1.3 Limits At Infinity; End Behavior Of A Function - Exercises Set 1.3 - Page 79: 31

$$0$$

Given $$\lim_{x\to \infty} (\sqrt{x^2+3}-x)$$ Then \begin{align*} \lim_{x\to \infty} (\sqrt{x^2+3}-x)&=\lim_{x\to \infty} (\sqrt{x^2+3}-x)\frac{\sqrt{x^2+3}+x}{\sqrt{x^2+3}+x}\\ &=\lim_{x\to \infty} \frac{x^2+3 -x^2}{\sqrt{x^2+3}+x}\\ &=\lim_{x\to \infty} \frac{ 3 }{\sqrt{x^2+3}+x}\\ &=\lim_{x\to \infty} \frac{ 3/x}{\sqrt{x^2/x^2+3/x^2}+x/x}\\ &=\lim_{x\to \infty} \frac{ 3/x}{\sqrt{1+3/x^2}+1}\\ &=\frac{0}{1}\\ &=0 \end{align*}