Chapter 2 - Limits - 2.5 Limits at Infinity - 2.5 Exercises - Page 97: 53

$$\eqalign{ & a.\,\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = 2;\,\,\,\mathop {\lim }\limits_{x \to - \infty } f\left( x \right) = 2;\,\,\,\,\,\,y = 2 \cr & b.\,x = 0;\,\,\,\,\,\mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right) = - \infty ;\,\,\,\,\,\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = + \infty \cr} $$

$$\eqalign{ & f\left( x \right) = \frac{{2{x^3} + 10{x^2} + 12x}}{{{x^3} + 2{x^2}}} \cr & {\text{By using the long division}} \cr & f\left( x \right) = 2 + \frac{{6{x^2} + 12x}}{{{x^3} + 2{x^2}}} \cr & f\left( x \right) = 2 + \frac{{6x + 12}}{{{x^2} + 2x}} \cr & {\text{Factor}} \cr & f\left( x \right) = 2 + \frac{{6\left( {x + 2} \right)}}{{x\left( {x + 2} \right)}} \cr & f\left( x \right) = 2 + \frac{6}{x} \cr & \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) \to \mathop {\lim }\limits_{x \to \infty } \left( {2 + \frac{6}{x}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \,\,\mathop {\lim }\limits_{x \to \infty } \left( 2 \right) + \overbrace {\,\mathop {\lim }\limits_{x \to \infty } \left( {\frac{6}{x}} \right)}^{{\text{approaches 0}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \,2 + 0 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \,2 \cr & \cr & \mathop {\lim }\limits_{x \to - \infty } f\left( x \right) \to \mathop {\lim }\limits_{x \to - \infty } \left( {2 + \frac{6}{x}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \,\,\mathop {\lim }\limits_{x \to - \infty } \left( 2 \right) + \overbrace {\,\mathop {\lim }\limits_{x \to - \infty } \left( {\frac{6}{x}} \right)}^{{\text{approaches 0}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \,2 + 0 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \,2 \cr & {\text{Then, the horizontal asymptote of }}f\left( x \right){\text{ is}} \cr & y = 2 \cr & \cr & b.{\text{ The function is not defined for }}x = 0 \cr & {\text{Then, there is a vertical asymptote at }}x = 0 \cr & \cr & {\text{Analyze }}\mathop {\lim }\limits_{x \to {0^ - }} \left( {2 + \frac{6}{x}} \right) \cr & \mathop {\lim }\limits_{x \to {0^ - }} \left( {2 + \frac{6}{x}} \right) = \mathop {\lim }\limits_{x \to {0^ - }} \left( 2 \right) + \overbrace {\mathop {\lim }\limits_{x \to {0^ - }} \left( {\frac{6}{x}} \right)}^{{\text{approaches - }}\infty } \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 2 - \infty \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \infty \cr & \cr & {\text{Analyze }}\mathop {\lim }\limits_{x \to {0^ + }} \left( {2 + \frac{6}{x}} \right) \cr & \mathop {\lim }\limits_{x \to {0^ - }} \left( {2 + \frac{6}{x}} \right) = \mathop {\lim }\limits_{x \to {0^ + }} \left( 2 \right) + \overbrace {\mathop {\lim }\limits_{x \to {0^ + }} \left( {\frac{6}{x}} \right)}^{{\text{approaches + }}\infty } \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 2 + \infty \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = + \infty \cr} $$