Chapter 2 - Limits - 2.5 Limits at Infinity - 2.5 Exercises - Page 97: 59

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

$$\eqalign{ & f\left( x \right) = \frac{{\sqrt {{x^2} + 2x + 6} - 3}}{{x - 1}} \cr & {\text{Rationalizing}} \cr & f\left( x \right) = \frac{{\sqrt {{x^2} + 2x + 6} - 3}}{{x - 1}} \times \frac{{\sqrt {{x^2} + 2x + 6} + 3}}{{\sqrt {{x^2} + 2x + 6} + 3}} \cr & f\left( x \right) = \frac{{{{\left( {\sqrt {{x^2} + 2x + 6} } \right)}^2} - 9}}{{\left( {x - 1} \right)\left( {\sqrt {{x^2} + 2x + 6} + 3} \right)}} \cr & f\left( x \right) = \frac{{{x^2} + 2x + 6 - 9}}{{\left( {x - 1} \right)\left( {\sqrt {{x^2} + 2x + 6} + 3} \right)}} \cr & f\left( x \right) = \frac{{{x^2} + 2x - 3}}{{\left( {x - 1} \right)\left( {\sqrt {{x^2} + 2x + 6} + 3} \right)}} \cr & f\left( x \right) = \frac{{\left( {x + 3} \right)\left( {x - 1} \right)}}{{\left( {x - 1} \right)\left( {\sqrt {{x^2} + 2x + 6} + 3} \right)}} \cr & f\left( x \right) = \frac{{x + 3}}{{\sqrt {{x^2} + 2x + 6} + 3}}, x\ne1 \cr & \cr & {\text{Divide the numerator and denominator by }}x \cr & f\left( x \right) = \frac{{\frac{x}{x} + \frac{3}{x}}}{{\sqrt {\frac{{{x^2}}}{{{x^2}}} + \frac{{2x}}{{{x^2}}} + \frac{6}{{{x^2}}}} + \frac{3}{x}}} \cr & f\left( x \right) = \frac{{1 + \frac{3}{x}}}{{\sqrt {1 + \frac{2}{x} + \frac{6}{{{x^2}}}} + \frac{3}{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) \cr & \,\,\,\,\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{1 + \frac{3}{x}}}{{\sqrt {1 + \frac{2}{x} + \frac{6}{{{x^2}}}} + \frac{3}{x}}}} \right) = \frac{{1 + \frac{3}{\infty }}}{{\sqrt {1 + \frac{2}{\infty } + \frac{6}{{{{\left( \infty \right)}^2}}}} + \frac{3}{\infty }}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{1 + 0}}{{\sqrt {1 + 0 + 0} + 0}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1 \cr & *\,\,\mathop {\lim }\limits_{x \to - \infty } f\left( x \right) \cr & \,\,\,\, - \mathop {\lim }\limits_{x \to - \infty } \left( {\frac{{1 + \frac{3}{x}}}{{\sqrt {1 + \frac{2}{x} + \frac{6}{{{x^2}}}} + \frac{3}{x}}}} \right) = - \frac{{1 + \frac{3}{\infty }}}{{\sqrt {1 + \frac{2}{\infty } + \frac{6}{{{{\left( \infty \right)}^2}}}} + \frac{3}{\infty }}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{1}{{1 - 0}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - 1 \cr & \cr & b.\,\,{\text{There is no vertical asymptote, because the denominator is}} \cr & {\text{always positive}}{\text{.}} \cr} $$