1. Question Suppose exists, find the limit: (1) (2) Sol. (1) (2)
(1) (2)
Sol. (1) (2)
(1) Suppose exists and then
(2) Suppose as then

2. Question
Suppose exists and find the limit
The solution is
Sol.

3. Derivatives of logarithmic functions
The derivative of is
Putting a=e, we obtain

4. Example
Ex. Differentiate
Sol.

5. Question
Find if
Sol. Since
it follows that
Thus for all

6. Example Find if Sol. Since it follows that and by definition,
Thus for all x

7. Question
Find if (a) (b)
(c)
Sol. (a)
(b)

8. The number e as a limit We have known that, if then
Thus, which by definition, means
Or, equivalently, we have the following important limit

9. Other forms of the important limit
Putting u=1/x, we have
More generally, if then

10. Question Suppose exists and find the limit The solution is
Sol. Let then

11. Question
Discuss the differentiability of
and find
Sol.
does not exist

12. Homework 6
Section 3.6: 46, 49, 50
Section 3.7: 16, 20, 34, 35, 39, 40, 63
Section 3.8: 41, 45, 48