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Unformatted text preview: 18.01 Calculus Jason Starr Fall 2005 Lecture 10. September 30, 2005 Homework. Problem Set 3 Part I: (a)(f). Part II: Problems 1, 2 and 3. Practice Problems. Course Reader: 2C-5, 2C-10, 2C-12, 2D-3, 2D-4. 1. Asymptotes. An asymptote describes the behavior of the graph of y = f ( x ) as it becomes unbounded, in some sense. There are two main examples. The function f has a vertical asymptote x = a if at least 1 of the following holds, lim f ( x ) = + , lim f ( x ) = , lim f ( x ) = + , lim f ( x ) = . x a x a x a + x a + In each case, the graph of y = f ( x ) becomes unbounded, and becomes arbitrarily close to the line x = a . If x = a is a vertical asymptote, then f ( x ) has an infinite discontinuity at x = a . The function f has a horizontal asymptote y = b if at least 1 of the following holds, lim f ( x ) = b, lim f ( x ) = b. x + x In other words, the graph of y = f ( x ) becomes arbitrarily close to the line y = b as x approaches either + or . 2 2 Example. For the function y = ( x 3 + x ) / ( x 1) = x ( x 2 + 1) / ( x 1), the lines x = 1 and x = 1 are vertical asymptotes. There is no horizontal asymptote. However, the graph of y is asymptotic to the line y = x . This was not discussed in lecture. A pair of functions f and g are asymptotic to each other if the line y = is a horizontal asymptote of f g ....
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