1. Infinite Limits Determine infinite limits from the left and from the right. Find and sketch the vertical asymptotes of the graph of a function.

2. Infinite Limits A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite limit. Stating that the limit of f(x) = ∞ as x  c does not mean that the limit exists. Infinity is not a number; it is used to denote the unbounded behavior of f(x) as x approaches c.

3. Asymptotes: Here is where limits are useful and not just a problem we calculate… we can use them to determine asymptotes, both horizontal and vertical

4. Vertical Asymptotes If f(x) approaches infinity (or negative infinity) as x approaches c from the right or left, then the line x = c is a vertical asymptote of the graph of f.

5. Vertical Asymptotes More formal: Let f and g be continuous on an open interval containing c. If f(c)≠0, g(c)=0, and there exists an open interval containing c such that g(x) ≠0 for all x≠c in the interval, then the graph of the function given by has a vertical asymptote at x=c

6. Finding VERTICAL asymptotes: where is this function undefined? What makes this function ‘blow up’? Set the denominator equal to 0 and solve!!!

7. Finding VERTICAL asymptotes: where is this function undefined?

8. Finding VERTICAL asymptotes: Rewrite this in terms of sine and cosine and then you can see where it is undefined.

9. Properties of Limits: Let c and L be real numbers and let f and g be functions such that and 1. Sum or difference: 2. Product: L>0 L<0 3. Quotient: This last one is important, we will revisit it in chapter 3, but the next slide gives you an idea.

10. Limits at Infinity or Horizontal Asymptotes:

11. Homework: P. 88 #1-28, 37-48 all

12. Note: Observe that the statement (under asymptotes) requires that you have only the denominator zero and not the numerator. If top and bottom are both zero, you have an indeterminate form and can’t determine the behavior at x = c without further investigation. Examples on p. 84-87