Limits

Find and explain Answer: Does not exist But why?!

Consider: What happens as x approaches zero? Numerically: You can scroll down to see more values.

Graphically:

Looks like y=1 It appears that the limit of as x approaches zero is 1

…gives us this window Turn the Axes off and then graph it again.

Looks like y=1 It again appears that the limit as x approaches zero is 1 even though it is not defined there.

Limit notation: “The limit of f of x as x approaches c is L.” So:

So what about this one: But remember that: …and so… 1 1 1

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

Properties of Limits: Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. (See page 58 for details.) For a limit to exist, the function must approach the same value from both sides. One-sided limits approach from either the left or right side only.

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

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

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

because at any endpoint, only a one-sided limit is necessary for the limit to exist. 2 1 1 2 3 4 At x=4: left hand limit DNE right hand limit value of the function

Find …with a hole at x = 2 And what does the graph look like?

The Sandwich Theorem: Show that: The maximum value of sine is 1, so The minimum value of sine is -1, so So:

By the sandwich theorem: WINDOW

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