Continuity and One Sided Limits

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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 f ( x)  DNE
x c

3. lim f ( x)  f (c)
x c

A function f is continuous at c IFF ALLare
true…
 1. f(c) is defined.
f ( x) exists
 2. lim
x c
 3. lim f ( x)  f (c)
x c

A function is continuous on an interval (a, b) if
it is continuous at each pt on the interval.

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:
x2  4
f ( x) 
x2
( x  2)( x  2)

x2
 x2
 We “removed” the (x-2).
▪ Therefore, we have a REMOVABLE DISCONTINUITY
when x – 2 = 0, or, when x = 2.

Non-Removable: f ( x)  1
x 1
 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. 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. lim 3  x
x 3
=0

2.
lim
x 0
lim
x 0
x
=1
x
x
= -1
x
Therefore, the limit as x approaches 0
DNE!!
4  x, x  1

 3. f ( x ) 

2
4
x

x
, x 1

lim f ( x)  3
x 1
lim f ( x) 
3
lim f ( x) 
3
x 1
x 1




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.
2 x  3, x  1
f ( x); f ( x)   2

 Ex: lim
x 1
x , x  1




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.
2 x  3, x  1
lim f ( x); f ( x)   2

x 1
x , x  1


lim  2 x  3
x 1
lim x
x 1
2
=1
=1
Therefore, the limit as x
approaches 1 of f(x) =1
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