1.3 The limit of a function. A rock falls from a high cliff. The position of the rock is given by: After 2 seconds: average speed: What is the instantaneous.

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Presentation transcript:

1.3 The limit of a function

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

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.

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

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!

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

At x=3:left hand limit right hand limit value of the function because the left and right hand limits match