How to solve this question

lim 7x2 - 6.x - 2 -+00 2x2 + 2x + 1 SOLUTION As x becomes large, both numerator and denominator become large, so it isn't obvious what happens to their ratio. We need to do some preliminary algebra. To evaluate the limit at infinity of any rational function, we first divide both numerator and denominator by the highest power of x that occurs in the denominator. (We may assume that x = 0, since we are interested only in large values of x.) In this case the highest power of x in the denominator is x, so we have Video Example () 7x2-6x-2 Tutorial lim 7x2 - 6.x - 2 x-+co 2x2 + 2x + 1 lim x-+00 2x-+2x+1 Online Textbook = lim T-+00 2 + + lim 6 1-+00 lim 2+ T-+00 HIN lim 7 - 6 lim _ 1-+00 2 lim I-+00 lim 2 + 2 lim - + lim 1+00 X I-+00 2:2 7 - 2 + + A simple calculation shows that the limit as x - -co is also . The figure illustrates the results of these calculations by showing how the graph of the given rational function approaches the horizontal asymptote y =EXAMPLE 3 Evaluate the limit below and indicate which properties of limits are used at each stage. lim 7.x2 - 6.x - 2 -+oo 2x2 + 2x +1 SOLUTION As x becomes large, both numerator and denominator become large, so it isn't obvious what happens to their ratio. We need to do some preliminary algebra. To evaluate the limit at infinity of any rational function, we first divide both numerator and denominator by the highest power of x that occurs in the denominator. (We may assume that x = 0, since we are interested only in large values of x.) In this case the highest power of x in the denominator is x, so we have Video Example () 7x2 - 6.x - 2 7x2-6x-2 lim Tutorial lim x-+oo 2x2 + 2x + 1 x-+00 2x-+2x+1 Online Textbook - lim 1-+00 2 + lim 6 1-+00 lim 2+ T-+00 HIN lim 7 - 6 lim _ 1-+00 2 lim I-+00 lim 2 + 2 lim - + lim 1+00 X I-+00 2:2 7 - 2 + + A simple calculation shows that the limit as x - -co is also . The figure illustrates the results of these calculations by showing how the graph of the given rational function