Download Limits - friendlymath

We will learn about:
limits, finding limits,
one-sided limits, and
unbound behavior of functions
The word “limit” is used in everyday
conversation to describe the ultimate behavior
of something, as in the “limit of one’s
endurance” or the “limit of one’s patience.”
In mathematics, the word “limit” has a similar
but more precise meaning.
Given a function f(x), if x approaching 3 causes
the function to take values approaching (or
equaling) some particular number, such as 10,
then we will call 10 the limit of the function
and write

Graphically – look at the picture/graph

Numerically – analyze the table of values

Analytically – algebraic techniques


Find the limit:
Draw graph: What is happening to y
as x gets closer to 1?

This table shows what
f (x) is doing as x approaches 3.
Or, we have the limit of the function as x approaches 3.
We write this procedure with the following notation.
lim  2 x  4   10
We write:
x3
10
lim f ( x)  L
General notation:
x c
Or, as x → c, then f
(x) → L
If the functional value of f
(x) is close to the single real number L
whenever x is close to, but not equal to, c (on either side of c).
x
2
2.9
2.99
2.999
3
3.001
3.01
3.1
4
f (x)
8
9.8
9.98
9.998
?
10.002
10.02
10.2
12
Find the limit:
x2 - 4
1. f ( x ) =
x -2
x2  4
lim
4
x 2 x  2
x-2
x2
2. f ( x ) =
lim
 Does Not Exist
x 2 x  2
x-2
ìx2 x ¹ 2
lim f  x   4
3. f ( x ) = í
x 2
0
x
=
2
î

Looking at what happens to f (x) when x
approaches c.


It doesn’t matter what happens AT c, just what
happens as x gets closer to c.
x must approach c from either side of c.
If the value when x approaches c from the left
does not equal the value when x approaches c
from the right, then…
The Limit Does Not Exist!!!!

If f (x) becomes arbitrarily close to a single
number L as x approaches c from either side,
then
This is read as
“the limit of f (x) as x approaches c is L.”

Let b and c be real numbers, and let n be a
positive integer.

Let b and c be real numbers, let n be a
positive integer, and let f and g be functions
with the following limits.
= -85
= -2
=4


0
0
is an indeterminate form.
Use the Replacement Theorem:
Let c be a real number and let f (x) = g (x) for all x
If the limit of g (x) exists as x approaches c, then
the limit of f (x) also exists and
¹c
x -4
lim
=?
x®2 x - 2
2
x 4
lim

x2 x  2
 x  2  x  2 
 lim
 lim  x  2   4
x 2
x 2
x2
2
x+4 -2
lim
=?
x®0
x
x42
lim

x0
x
x42
 lim

x0
x
 lim
x0
x
 lim
x0
x




x  44
x42
x
x42
1

x  4  2
x42


1
 lim

x0
x42 4

 x  2 x  1
x2 3
 lim
 lim

x 1  x  1 x  1
x 1 x  1
2

 lim
xh x
xh x

h 0
h
xh x
x  h x
h
 lim
 lim
h 0
h 0
h xh x
h xh x

 lim
h 0


1
xh x



1
2 x


While evaluating limits from graphs or tables:
x
2
2.9
2.99
2.999
3
3.001
3.01
3.1
4
f (x)
8
9.8
9.98
9.998
?
10.002
10.02
10.2
12
We saw: the numbers 2.9, 2.99, 2.999, ...
approach 3 from the left, which we denote by
x→3 –, and the numbers 3.1, 3.01, 3.001, ...
approach 3 from the right, denoted by x→3 +.
Such limits are called one-sided limits.

Limit from the Left
lim f  x   L
x c

Limit from the Right
lim f  x   L
x c
= -1
=1
lim f  x   1
x 0
If f (x) is a function and c and L are real
numbers, then
lim f  x   L
x c
if and only if
both the left and right limits exist
and are equal to L

Another way for the limit to fail to exist:
1
lim
 
x3 x  3
1
lim

x3 x  3