1. 3.7 Graphing Rational Functions Obj: graph rational functions with asymptotes and holes and evaluate limits of rational functions

2. Graph f(x) = g(x) = Infinite versus Removable Discontinuity (asymptote) (point)

3. Definitions Let y = y has an infinite discontinuity at x = a if n > m (because the factors cancelled, and there are only factors left over in the denominator) y has a removable (point) discontinuity at x = a if m > n (because the factors cancelled, and there are only factors left over in the numerator)

4. Examples f(x) = Factored: Since x – 3 cancels in the numerator and denominator,the graph has removable discontinuity at x = 3.

5. Theorem Let h be a rational function, If h has an infinite discontinuity at x = a, then the graph of h has a vertical asymptote at x = a. If h has a removable discontinuity at x = a, then the graph of h has a hole at x = a.

6. Horizontal Asymptotes Use limits to find any horizontal asymptotes of the graph of each function. To find limits as x approaches infinity, substitute ∞ for x, and evaluate.

7. Practice Calculate each limit.

8. Find Asymptotes Find the vertical asymptote(s) and horizontal asymptote, if any. 1) Factor, cancel any like factors. 2) To find vertical asymptotes, set denominator = 0, solve for x. 3) To find horizontal asymptotes, find the limit as x approaches infinity. Does the function have any holes?

9. Sketch a graph Sketch the graph using the vertical and horizontal asymptotes. f(x)= 1) Factor and cancel to find holes 2) Locate asymptotes 3) Locate x- and y-intercepts 4) Graph

10. Your Turn Round Table Practice Assignment page 215 - 217 10 – 12, 15, 18, 19a, b, 22a, b, c