1.6 One-Sided Limits and Continuity

One Sided Limit

Limit as approach a value from only one direction

Definition – One-Sided Limits

If f(x) approaches L as x tends toward c from the left (x<c),

then

Likewise, if f(x) approaches M as x tends toward c from the right (c<x), then

Example

#24 Determine right and left sided limits for the function at x = 0

#6 #12

Existence of a Limit

The two-sided limit exists if and only if the two one-sided limits and both exist and are equal, then

Example

#24 Decide if

Continuity

Informally, can draw without lifting pencil.

Definition

A function f is continuous at c if all three of these conditions are satisfied:

If f(x) is not continuous at c, it is said to have a discontinuity there.

A polynomial or rational function is continuous wherever it is defined.

Example

Test continuity using the definition of continuity

General Continuity Properties

If two functions are continuous on the same interval, then their sum, difference, product, and quotient are continuous on the same interval, except for values of x that make a denominator 0.

Examples

List all values where the function is not continuous?

#32 #36

#38 #40

Application: Look in text (TRANSPARENCY) – page 88 - #48

#54 Find the value of the constant A such the function f(x) will be continuous for all values of x

The Intermediate Value Property

If f(x) is continuous on the interval a<x< b and L is a number between f(a) and f(b), then f(c) = L for some number c between a and b.

That is: A continuous function attains all values between any two of its values.

A new born is 21” long and 3’ (or 36”) tall as a child – then at some age the child was 30” tall.

#56

Show that the equation has at least one solution for the interval 0 < x < 1

Assignment: 1.6; pg 86; 1-47 odd, 53, 57