Limit of a Composite Function

If f and g are functions such that and , then

lim ( )x cg x L

lim ( ) ( )x L

f x f L

lim ( ( )) lim ( ) ( ).x c x cf g x f g x f L

Example

Because and

it follows that

2

0lim( 4) 4x

x

4

lim 2x

x

2

0lim 4 4 2x

x

Definition of Continuity

Continuity at a Point: A function f is continuous at c if the following three conditions are met.

1. f(c) is defined.

2. 3.

lim ( ) exists.x cf x

lim ( ) ( ).x cf x f c

Continuity

Continuity on an Open Interval:A function is continuous on an open interval (a,b) if it

is continuous at each point on the interval.

A function that is continuous on the entire real line is everywhere continuous.

Continuity

Continuity on a Closed Interval:A function f is continuous on a closed interval [a,b] if it

is continuous on the open interval (a,b) and

lim ( ) ( ) and lim ( ) ( ).x a x b

f x f a f x f b

Intermediate Value Theorem

If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k.