Chapter 2 - Limits - 2.5 Limits at Infinity - 2.5 Exercises - Page 97: 58

$$\eqalign{ & a.{\text{No horizontal asymptotes}} \cr & b.\,x = - 1;\,\,\,\,\,\mathop {\lim }\limits_{x \to - {1^ - }} f\left( x \right) = - \infty ;\,\,\,\,\,\mathop {\lim }\limits_{x \to - {1^ + }} f\left( x \right) = + \infty \cr} $$

$$\eqalign{ & f\left( x \right) = \frac{{x - 1}}{{{x^{2/3}} - 1}} \cr & a.{\text{ Calculate }}\mathop {\lim }\limits_{x \to \infty } f\left( x \right){\text{ and }}\mathop {\lim }\limits_{x \to - \infty } f\left( x \right) \cr & *\,\,\mathop {\lim }\limits_{x \to \infty } f\left( x \right) \cr & \,\,\,\,\mathop {\lim }\limits_{x \to \infty } \frac{{x - 1}}{{{x^{2/3}} - 1}} = \frac{{\infty - 1}}{{{\infty ^{2/3}} - 1}} = \frac{\infty }{\infty } \cr & {\text{Divide the numerator and denominator by }}x \cr & \,\,\,\,\mathop {\lim }\limits_{x \to \infty } \frac{{x - 1}}{{{x^{2/3}} - 1}} = \,\,\,\mathop {\lim }\limits_{x \to \infty } \frac{{\frac{x}{x} - \frac{1}{x}}}{{\frac{{{x^{2/3}}}}{x} - \frac{1}{x}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \,\,\,\mathop {\lim }\limits_{x \to \infty } \frac{{1 - \frac{1}{x}}}{{\frac{1}{{{x^{1/3}}}} - \frac{1}{x}}} \cr & {\text{Using limits properties}} \cr & = \,\frac{{\mathop {\lim }\limits_{x \to \infty } \left( 1 \right) - \overbrace {\mathop {\lim }\limits_{x \to \infty } \left( {\frac{1}{x}} \right)}^{{\text{approaches 0}}}}}{{\underbrace {\mathop {\lim }\limits_{x \to \infty } \left( {\frac{1}{{{x^{1/3}}}}} \right)}_{{\text{approaches 0}}} - \underbrace {\mathop {\lim }\limits_{x \to \infty } \left( {\frac{1}{x}} \right)}_{{\text{approaches 0}}}}} \cr & = \frac{{1 - 0}}{0} = + \infty \cr & and\,similarly \cr & \,\mathop {\lim }\limits_{x \to - \infty } = - \infty \cr & {\text{Then}}{\text{, there are no horizontal asymptotes}}{\text{.}} \cr & \cr & b.\, \cr & f\left( x \right) = \frac{{x - 1}}{{{x^{2/3}} - 1}} \cr & {\text{The function is not defined for }}x = - 1,{\text{ then there is a}} \cr & {\text{vertical asymptote at }}x = - 1 \cr & \cr & {\text{Analyze }}\mathop {\lim }\limits_{x \to - {1^ - }} f\left( x \right){\text{ and }}\mathop {\lim }\limits_{x \to - {1^ + }} f\left( x \right) \cr & f\left( x \right) = \frac{{x - 1}}{{{x^{2/3}} - 1}} = \frac{{{x^{2/3}} + {x^{1/3}} + 1}}{{{x^{1/3}} + 1}} \cr & \mathop {\lim }\limits_{x \to - {1^ - }} f\left( x \right) = \mathop {\lim }\limits_{x \to - {1^ - }} \left( {\frac{{{x^{2/3}} + {x^{1/3}} + 1}}{{{x^{1/3}} + 1}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{{\left( { - {1^ - }} \right)}^{2/3}} + {{\left( { - {1^ - }} \right)}^{1/3}} + 1}}{{{{\left( { - {1^ - }} \right)}^{1/3}} + 1}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{{{0^ - }}} = - \infty \cr & \mathop {\lim }\limits_{x \to - {1^ + }} f\left( x \right) = \mathop {\lim }\limits_{x \to - {1^ + }} \left( {\frac{{{x^{2/3}} + {x^{1/3}} + 1}}{{{x^{1/3}} + 1}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{{\left( { - {1^ + }} \right)}^{2/3}} + {{\left( { - {1^ + }} \right)}^{1/3}} + 1}}{{{{\left( { - {1^ - }} \right)}^{1/3}} + 1}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{3}{{{0^ + }}} = + \infty \cr} $$