Chapter 2: Limits and Continuity - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises 2.6 - Page 97: 15

(a) $0$ (b) $0$

(a) Since \begin{align*} \lim_{x\to \infty }f(x) &=\lim_{x\to \infty }\frac{x+1}{x^2+1}\\ &=\lim_{x\to \infty }\frac{x/x^2+1/x^2}{x^2/x^2+1/x^2}\\ &=\frac{\lim_{x\to \infty }(1/x)+\lim_{x\to \infty }(1/x^2)}{\lim_{x\to \infty }(1)+\lim_{x\to \infty }(1/x^2)}\\ &=\frac{0+0}{1+0}\\ &=0 \end{align*} (b) Since \begin{align*} \lim_{x\to-\infty }f(x) &=\lim_{x\to- \infty }\frac{x+1}{x^2+1}\\ &=\lim_{x\to -\infty }\frac{x/x^2+1/x^2}{x^2/x^2+1/x^2}\\ &=\frac{\lim_{x\to -\infty }(1/x)+\lim_{x\to- \infty }(1/x^2)}{\lim_{x\to- \infty }(1)+\lim_{x\to- \infty }(1/x^2)}\\ &=\frac{0+0}{1+0}\\ &=0 \end{align*}