1.3 The limit of a function

1. 1.3 The limit of a function

2. A motivating example 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?

3. 1 80 0.1 65.6 .01 64.16 .001 64.016 64.0016 .0001 .00001 64.0002 We can use a calculator to evaluate this expression for smaller and smaller values of h. 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.)

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. does not exist because the left and right hand limits do not match! left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=1:

9. because the left and right hand limits match. left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=2:

10. because the left and right hand limits match. left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=3: