Lehigh University Math 21 - Limits and Continuity

One Sided Limits

For the function

f(x) = ì
í
î

3x+1,

if x ³ 1

x+2,

if x < 1

,

as far as the previous definition goes, there is no limit. As you approach x = 1 from the left ( x near 1 but x < 1 ), the value of f(x) is approaching 3 (following the red part of the graph), but if you approach from the right ( x near 1 but x > 1 ), the value of f(x) is approaching 4 , following the blue part of the graph. There really are limits at 1, as long as you don't approach 1 from both sides. That's called a one-sided limit:

Definition 1 We say that the limit from the left (or below) of f(x) at a exists and equals L , and write

lim
x® a-
f(x) = L æ
è or
lim
x a
f(x) = L ö
ø ,

if we can make f(x) as close as we need to L just by taking x close enough to a , keeping x < a .

You can imagine how we define

lim
x® a+
f(x) = L æ
è or
lim
x¯ a
f(x) = L ö
ø .

In order to have a real limit at a , it is enough to have both one sided limits as x approaches a from above and from below, and the two one-sided limits have to be the same. That is:

Theorem 2

lim
x® a
f(x) = L Û
lim
x® a+
f(x) = L and
lim
x® a-
f(x) = L.

Example 1 Find

lim
x 1
x²-1
| x-1|
.

How to enter math formulas

Exercise 1 If

f(x) = ì
í
î

3x²+x,

if x ³ 1

4x,

if x < 1

,

find all of the following that exist:

lim
x 1
f(x) =

lim
x¯ 1
f(x) =

lim
x® 1
f(x) =

Exercise 2 If

f(x) = ì
í
î

3x²+x,

if x > 2

4x,

if x £ 2

,

find all of the following that exist:

lim
x 2
f(x) =

lim
x¯ 2
f(x) =

lim
x® 2
f(x) =

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