Chapter 2: Limits and Continuity - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises 2.6 - Page 97: 17

(a) $7$ (b) $7$

(a) \begin{align*} \lim_{x\to \infty}h(x)&= \lim_{x\to \infty} \frac{ 7x^3}{x^3-3x^2 +6x}\\ &= \lim_{x\to \infty} \frac{ 7x^3/x^3}{x^3/x^3-3x^2/x^3 +6x/x^3}\\ &=\frac{ \lim_{x\to \infty}(7)}{ \lim_{x\to \infty}(1)- \lim_{x\to \infty}(3/x) + \lim_{x\to \infty}(6/x^2)}\\ &=\frac{7}{1-0+0}\\ &=7 \end{align*} (b)\begin{align*} \lim_{x\to -\infty}h(x)&= \lim_{x\to- \infty} \frac{ 7x^3}{x^3-3x^2 +6x}\\ &= \lim_{x\to -\infty} \frac{ 7x^3/x^3}{x^3/x^3-3x^2/x^3 +6x/x^3}\\ &=\frac{ \lim_{x\to -\infty}(7)}{ \lim_{x\to -\infty}(1)- \lim_{x\to- \infty}(3/x) + \lim_{x\to- \infty}(6/x^2)}\\ &=\frac{7}{1-0+0}\\ &=7 \end{align*}