1. 1.3 The limit of a function

2. A rock falls from a high cliff. The position of the rock is given by: After 2 seconds: average speed: What is the instantaneous speed at 2 seconds? A motivating example

3. 1 80 0.165.6.0164.16.00164.016.0001 64.0016.0000164.0002 We can see that the velocity approaches 64 ft/sec as h becomes very small. We say that the velocity has a limiting value of 64 as h approaches zero. (Note that h never actually becomes zero.) We can use a calculator to evaluate this expression for smaller and smaller values of h.

4. Definition of Limit We write and say “the limit of f(x), as x approaches a, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a. In our example,

5. The limit of a function refers to the value that the function approaches, not the actual value (if any). not 1

6. Left-hand and right-hand limits We write and say the left-hand limit of f(x) as x approaches a is equal to L if we can make the values of f(x) arbitrarily close to to L by taking x to be sufficiently close to a and x less than a. Similarly, we write and say the right-hand limit of f(x) as x approaches a is equal to L if we can make the values of f(x) arbitrarily close to to L by taking x to be sufficiently close to a and x greater than a.

7. Note that if and only if and

8. 1234 1 2 At x=1:left hand limit right hand limit value of the function does not exist because the left and right hand limits do not match!

9. At x=2:left hand limit right hand limit value of the function because the left and right hand limits match. 1234 1 2

10. At x=3:left hand limit right hand limit value of the function because the left and right hand limits match. 1234 1 2