1. Example Ex. Find all asymptotes of the curve Sol. So x=3 and x=-1 are vertical asymptotes. So y=x+2 is a slant asymptote.

2. Example Ex. Find asymptotes of the curve Sol. vertical asymptote horizontal asymptote Ex. Find asymptotes of the curve Sol. vertical asymptote slant asymptote

3. Curve sketching A. Domain B. Intercepts C. Symmetry D. Asymptotes E. Intervals of increase or decrease F. Local maximum and minimum values G. Convexity and points of inflection H. Sketch the curve

4. Example Ex. Sketch the graph of Sol. A. The domain is (- 1,+ 1 ). B. The y-intercept is 1. C. f is even. D. asymptotes: y=0 is horizontal asymptote. E. when x>0, so f(x) decreasing in (0,+ 1 ) and increasing in (- 1,0). F. x=0 is local and global maximum point. G. f(x) concave in and convex otherwise

5. Example Sketch the graph of

6. Example Ex. Prove the inequality: Proof. Let then

7. Indeterminate forms Question: find the limit we can’t apply the limit law because the limit of the denominator is 0. In fact the limit of the numerator is also 0. We call this type of limit an indeterminate form. Generally, if both and as then the limit may or may not exist and is called an indeterminate form of type 0/0

8. Previous methods For rational functions, for example, The important limit: Does not work for general cases. There is a systematic method, known as L’Hospital’s Rule, for evaluation of indeterminate forms.

9. L’Hospital’s rule L’Hospital’s Rule Suppose f and g are differentiable and near a (except possibly at a). Assume that and or that and Then if the last limit exists (can be a real number or or ).

10. Remarks Remark1. L’Hospital’s Rule can be used to evaluate the indefinite limit of type 0/0 or 1 / 1. Remark2. L’Hospital’s Rule is also valid when “x ! a” is replaced by x ! a+, x ! a-, x ! + 1, x ! - 1. Remark3. If is still an indeterminate type, we can use L’Hospital’s Rule again.

11. Examples Ex. Find Sol. Ex. Find Sol. Note: This example indicates that exponential infinity is much bigger than any power infinity.

12. When not to use L’Hospital’s rule Ex. Find Sol. L’Hospital’s Rule gives nothing! Correct solution is 0.

13. Other indeterminate types There are some other indeterminate types which can be changed into 0/0 or 1 / 1 type: 0 ¢1, 1 - 1, 1 1, 1 0, 0 0. Ex. Find Sol.

14. Homework 10 Section 4.4: 22, 30, 31, 45, 48, 55, 74 Section 4.5: 26, 28, 64