2.1 Introduction to Limits 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?

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. 1 80 0.1 65.6 .01 64.16 .001 64.016 .0001 64.0016 .00001 64.0002 We say that the velocity has a limiting value of 64 as h approaches zero. (Note that h never actually becomes zero.)

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,

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

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.

Note that if and only if and

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

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

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

Properties of Limits Suppose that c is a constant and the limits and exist. Then

Properties of Limits (cont.) Suppose that c is a constant, n is a positive integer and the limit exists. Then

Direct Substitution Property If f is a polynomial or a rational function and a is in the domain of f, then Examples on the board.

Indeterminate forms Consider: If we try to evaluate by direct substitution, we get: Zero divided by zero can not be evaluated. The limit may or may not exist, and is called an 0/0 indeterminate form. We can evaluate it by factoring and canceling: