continuity and one sided limits
DESCRIPTION
Section 2.4. Continuity and One Sided Limits. Continuity. 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. Breaking Continuity. 1. The function is undefined at x = c. Breaking Continuity. 2. - PowerPoint PPT PresentationTRANSCRIPT
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