Download Limits explained

LIMITS EXPLAINED
STUDYING THE BEHAVIOR OF FUNCTIONS
THERE ARE BASICALLY TWO TYPES OF LIMITS THAT GO INTO FINDING A LIMIT.

A left hand limit is denoted like this :
with the little uppercase negative behind a.

A right hand limit is denoted like this :
with the little uppercase positive behind a.
lim
xa
f (x)

lim
f (x)
xa
Both of these must match and be the same number in order to find:
lim
xa
f (x)
(Notice there is no “+” or “-“ behind the “a”.)
If the left hand limit and the right hand limit do not match and are not the same
number then
lim
xa
f (x)
does not exist.

lim
x  a
f (x)
is the y-value on the y-axis that the function gets close to as x gets close
to a from the left hand side of a on the x axis. In other words, as approaches 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 = 2 in the
picture)
f (x)
Consider
lim
x  2
f (x)
. This would mean we want to find the y – value that the function
gets close to as x approaches 2 from the left hand side of 2 on the x axis.
(Use your finger to follow the function as the value of x get closer to x = 2 on the x axis.)
That would be y = 1. So,
lim
x2
f (x)

= 1.

lim
xa
f (x)
is the y-value on the y-axis that the function gets close to as x gets
close to 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).
f (x)
So, consider
lim
x  2
f (x) .
Follow with your finger the function as you let x approach 2
from the right side of 2, and you approach the y – value y = 1 also.
Therefore,
lim
x  2
f (x)
= 1 as well.

So for the graph below, as you approach x = 2 from the left or from the right side, the
function value (y – values) approaches y = 1.
f (x)

Therefore, we then can say that the limit as x approaches 2 regardless of direction,
denoted is:
lim
x2
f (x)
= 1.
POINTS TO REMEMBER

If,
lim
xa

f (x)
=
lim
xa
f (x)
, then
lim
xa
f (x)
= that y- value!!!
f (x)
does not exist!!!
-AND
If,
lim
x  a
f (x)

lim
xa
f (x)
, then
lim
xa
Let’s look at an example of this second possibility. We will use the same graph as
before.

Using this same function, let’s find the following:
f (x)
lim
x  0
f (x)
= 2 since the y – value that the function approaches as x
approaches 0 from the left is y = 2.
lim
f (x)
x  0
= -1 since the y – value that the functions approaches as x
approaches 0 from the rights is y = -1.
Since the left hand limit and the right hand limit are different, we say
that:
lim
f (x)
x0
does not exist.
TO EVALUATE LIMITS WITHOUT DRAWING
PICTURES, HERE ARE THE STEPS YOU TAKE:

Step 1: Simply plug the value of x your 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.
Example :
Example:
lim
(3x  5)  3  4  5  12  5  7 ,
x4
lim
x  3
7  x 7  3 10


 10
4  x 4  3 1
so 7 is the answer.
, so 10 is the answer.

Step 2: If you get 0 , when you plug in the value of x, you 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.
0
lim ( x  1)( x  1)
lim
x 2 1
( x  1)  1  1  2


x 1
x  1 x 1
x 1
x 1
lim
Example :
, so 2 is the answer.
Notice in step 2 that I cancelled out (x – 1). Also, notice that the “limit”
symbol was carried from step to step until I was able to actually plug in the
x = 1 and get a value.
Example:
lim
x4
x 2
x4
=
lim
x4
( x  2) ( x  2)

( x  4) ( x  2)
=
lim
x4
x  4 ( x  4)( x  2)
=
lim
x4
1
x 2

1
4 2

1
1

22 4
So, ¼ is the answer. Notice in step 2 the top portion was “FOILed” but not the
bottom. I then canceled out (x – 4), in step 3. Again the “limit” symbol was carried
from step to step until I was able to actually plug in the x = 4 and get a value.
nonzero

Step 3: If you get 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: To find
lim
x2
5
x2

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 place of x in
5
x2
using a
calculator and you will get these numbers: -5,-10, -50,-500. As you can tell the
numbers are getting more and more negative so
lim
x2
To find
lim
x2

5
x2

5
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 the fraction gives 5, +10, +50,+500, so
lim
x2

5
x2
=


Since ∞ ≠ - ∞, we know that
lim
x2
5
x2
does not exist!!
(notice there is no “+” or “-“ behind 2).