Download Limits Explained There are basically two types of limits that go into

Limits Explained
There are basically two types of limits that go into finding a limit.
lim
A left hand limit is denoted like this
x  a
1. A right hand limit is denoted like this
f (x) with the little uppercase negative behind a.
lim
xa
f (x) with the little uppercase positive
behind a.
Both of these must match and be the same number in order to find
match and are not the same number then
lim
xa
lim
xa
f (x) . If they do not
f (x) does not exist. (Notice there is no “+” or “-
“ behind the a.)
Here are some things to know:
lim
f (x) is the y-value on the y-axis that the function f (x) gets close to as x gets close to a
x  a
from the left hand side of a. In other words, as x approaches a from the left, the function
approaches some y-value from the left. (Use your finger to follow what the function is doing on
the left hand side of a in the picture)
-ANDlim
f (x) is the y-value on the y-axis that the function f (x) gets close to as x gets close to a
xa
from the right hand side of a. In other words, as x approaches a from the right, the function
approaches some y-value from the right. (Again, use your fingers to follow the function)
-ANDif
if
lim
xa

lim
xa

f (x) =
f (x) 
lim
xa
lim
xa
f (x) then,
f (x) then
lim
xa
lim
xa
f (x) = that y-value.
f (x) does not exist!!
To evaluate limits without drawing pictures, here are the steps you take:
1. Simply plug the value of x you are approaching into the place of x. Do the arithmetic. If
you get a defined number, even a decimal or fraction, then that number is the limit value
and you are finished.
Examples:
lim
(3x  5)  3  4  5  12  5  7 , so 7 is the answer.
x4
7  x 7  3 10


 10 , so 10 is the answer.
x  3 4  x 4  3 1
0
If you get when you plug in the value of x, you
0
lim
2.
must do some algebra. Try factoring
and canceling, rationalizing and canceling, or simplifying and canceling. After
canceling, plug the value of x in again and you should get a number. That number will be
the limit.
Examples:
( x  1)( x  1) lim
x 2  1 lim

( x  1)  1  1  2 , so 2 is the answer. Notice in

x 1
x  1 x 1
x 1
x 1
lim
step 2 that I cancelled out ( x  1) .
( x  2) ( x  2)
lim
lim
lim
x 2
x4
=
=
=

x4
x  4 ( x  4) ( x  2)
x  4 ( x  4)( x  2)
x4
lim
x4
1
x 2
1

4 2

1
1
 ,
22 4
1
is the answer. Notice in step 2 the top portion was “FOILed” but not the bottom. I
4
then canceled out ( x  4) in step 3!!
so
NOTE: You stop writing the “lim” symbol when you can successfully plug in the
number for x and get a number.
3. If you get
nonzero
,
0
you must either draw the picture in your graphing calculator, if you
have one, or do what I like to call PLUG-N-CHUG.
Example:
lim
5
x

2
x2
To find

gives you
5
0
when you plug in x = 2. So use your calculator to plug
in numbers that get close to 2 coming from the left. That would be x = 1, x = 1.5, x = 1.9, x
= 1.99, etc. (These numbers steadily get closer to 2 from the left). Plug them each into the
5
x2
place of x in
and you will get these numbers: -5,-10, -50,-500. As you can tell the
numbers are getting more and more negative so,
To find
lim
x2

5
x2
lim
5
x

2
x2

=  .
, you will get close to 2 from the right with x = 3, x = 2.5, x = 2.1, x
= 2.01.
Plugging these numbers into
so
lim
x2

5
x2
will give you these numbers: 5, 10, 50, 500,
= .
Since     , we know
2).
5
x2
lim
5
does not exist (notice there is no “+” or “-“ behind
x 2 x2