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Unformatted text preview: 6. Limits Involving Infinity P. K. Lamm 09/13/11 (19:39) p. 1 / 10 Lecture Notes: 6. Limits Involving Infinity These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. In this set of notes well give more specific information about a number of limits that, in the past, we would have simply classified being undefined. 1. The Behavior of Polynomials at Infinity First we make some observations about polynomials: What happens to the curve y = x as x ? lim x - x =- , lim x + x = + . What happens to the curve y = x 2 as x ? lim x - x 2 = + , lim x + x 2 = + . In fact, we know that the curve y = x n for n 1 integer behaves as follows: lim x - x n = ( + , if n even ,- , if n odd , and lim x + x n = + . Now consider a general polynomial of degree n p ( x ) = a n x n + a n- 1 x n- 1 + a n- 2 x n- 2 + + a 1 x + a , where we assume that p is of degree n (i.e., the highest power of x appearing in the polynomial is n ); in this case it must be that a n 6 = 0. Since p ( x ) = a n x n 1 + a n- 1 a n 1 x + a n- 1 a n 1 x 2 + + + a 1 a n 1 x n- 1 + a a n 1 x n , where lim x 1 + a n- 1 a n 1 x + a n- 1 a n 1 x 2 + + + a 1 a n 1 x n- 1 + a a n 1 x n = 1 , this suggests that the limiting behavior of p ( x ) as x is the same as the limiting behavior of a n x n as x since a n x n is the dominant term in the polynomial. Example 1.1: lim x - 5 x 5- 21 x 2- 3 x- 1 =- . lim x 5 x 5- 21 x 2- 3 x- 1 = + . 1 6. Limits Involving Infinity P. K. Lamm 09/13/11 (19:39) p. 2 / 10 Example 1.2: lim x -- 3 x 18 + 20000 x 17 =- . lim x - 3 x 18 + 20000 x 17 =- . 2. The Behavior of Rational Functions at Infinity We now consider the limiting behavior of rational functions, i.e., of f ( x ) = p ( x ) q ( x ) where p ( x ) and q ( x ) are polynomials. Method: To evaluate lim x f ( x ) for rational functions, divide the numerator and denominator by the largest power of x in the denominator and then evaluate the limit. Example 2.1: Evaluate: lim x 2 x 2 + x x 2- 3 x + 1 . The largest power of x in the denominator is 2, so divide numerator and denominator by x 2 : lim x 2 x 2 + x x 2- 3 x + 1 1 x 2 1 x 2 = lim x 2 + 1 x 1- 3 x + 1 x 2 = 2 . Its worth nothing that the same result is obtained if we had taken the limit as x - . Example 2.2: Evaluate: lim x - x x 2- 3 x + 1 ....
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