Continuity and One Sided Limits

Section 2.4

To say a function is continuous at x = c means that there is NO interruption in the graph of f at c. The graph has no holes, gaps, or jumps.

1. The function is undefined at x = c

2. lim ( )x cf x DNE

3. lim ( ) ( )x cf x f c

A function f is continuous at c IFF ALLare true…

1. f(c) is defined. 2. 3.

A function is continuous on an interval (a, b) if it is continuous at each pt on the interval.

lim ( ) x cf x exists

lim ( ) ( )x cf x f c

A function is discontinuous at c if f is defined on (a, b) containing c (except maybe at c) and f is not continuous at c.

1. Removable : You can factor/cancel out, therefore

making it continuous by redefining f(c).

2. Non-Removable: You can’t remove it/cancel it out!

1. Removable:

We “removed” the (x-2).▪ Therefore, we have a REMOVABLE

DISCONTINUITY when x – 2 = 0, or, when x = 2.

2 4( )

2

xf x

x

( 2)( 2)

2

x x

x

2x

Non-Removable:

We can’t remove/cancel out this discontinuity, so we have a NON-Removable discontinuity when x – 1 =0, or when x = 1.▪ We will learn that Non-Removable

Discontinuities are actually Vertical Asymptotes!

1( )

1f x

x

1. Set the deno = 0 and solve. 2. If you can factor and cancel out

(ie-remove it) you have a REMOVABLE Discontinuity.

3. If not, you have a NON-Removable Discontinuity.

You can evaluate limits for the left side, or from the right side.

x approaches c from values that are greater than c.

x approaches c from values that are less than c.

1. 3

lim 3x

x

= 0

2.

0limx

x

x

0limx

x

x

= 1

= -1

Therefore, the limit as x approaches 0 DNE!!

3. 2

1

1

1

4 , 1( )

4 , 1

lim ( )

lim ( )

lim ( )

x

x

x

x xf x

x x x

f x

f x

f x

3

3

3

1. Factor and cancel as usual. 2. Evaluate the resulting function for

the value when x=c.3. If this answer is NOT UNDEFINED

then that is your solution. 4. If this answer is UNDEFINED, then

graph the function and look at the graph for when x=c.

Ex:

Evaluate each function separately for the value when x=c.

If the solutions are all the same, that is your limit.

If they are not, then the limit DNE.

21

2 3, 1lim ( ); ( )

, 1x

x xf x f x

x x

21

2 3, 1lim ( ); ( )

, 1x

x xf x f x

x x

1lim 2 3x

x

2

1limx

x

= 1

= 1

Therefore, the limit as x approaches 1

of f(x) =1