10.3 Limits
Objective:

Find the limit of a function using tables, graphs, and algebraic methods
WARMUP
(1) Use the TABLE feature of your calculator to fill in the table below for the function
x2  9
f ( x) 
x3
x
f ( x)
-3.1
-3.01
-3.001
-3
-2.999
-2.99
-2.9
From the table, we can conclude that the limit of this function as x approaches -3 is
_____.
(2) Repeat this process for the function f ( x)  3x 2  1
x
f ( x)
1.9
1.99
1.999
2
2.001
2.01
2.1
From the table, we can conclude that the limit of this function as x approaches 2 is
_____.
In both problems you were finding a limit using a table of values for x. What made the
process different in the second problem? Is there a quicker way to do #2?
Limits Using Tables and Graphs
Ex 1: Connecting the graph of a function to its limit
Graph f ( x) 
sin x
  
in your calculator in the window X :   ,  Y :[1, 2] on the plane
x
 4 4
below.
sin x

x 0
x
lim
Ex 2: Connecting the graph of a function to its limit
Graph f ( x) 
1  cos x
  
in your calculator in the window X :   ,  Y :[1, 2] on the
x
 4 4
plane below.
1  cos x

x 0
x
lim
Ex 3:
Graph f ( x ) 
1
in your calculator in the window X :  2, 2 Y :[2, 2] on the plane
x
below.
1

x 0 x
lim
Ex 4:
Ex 5:
lim f ( x) 
lim f ( x) 
x2
x 1
f ( x)
So….what is the difference between lim
x#
and f (#) ???
lim f ( x) 
x 1
f (1) 
Ex 6:
lim f ( x) 
x4
f (4) 
One-Sided Limit Notation
lim f ( x ) denotes a limit taken ONLY FROM THE LEFT OF THE NUMBER GIVEN (a)
xa
lim f ( x) denotes a limit taken ONLY FROM THE RIGHT OF THE NUMBER GIVEN (a)
xa
Ex 7:
lim f ( x) 
x 2
lim f ( x) 
x 3
lim f ( x) 
x  0
lim f ( x) 
x2
lim f ( x) 
x 3
lim f ( x) 
x 3
lim f ( x) 
x 0
lim f ( x) 
x  0
lim f ( x) 
x 3
Ex 8:
lim f ( x) 
x 2
lim f ( x) 
x  0
lim f ( x) 
x 3
lim f ( x) 
lim f ( x) 
lim f ( x) 
x 2
x  0
x 3
lim f ( x) 
lim f ( x) 
lim f ( x) 
f (2) 
f (0) 
f (3) 
x 2
x 0
x 3
What
can one
conclude about the existence of a limit of a function and the
graph of the function?
Finding Limits Algebraically
We have already seen that, in some cases, limits can be found by directly substituting a
value into a function, like the examples below.
Ex 9:
(a) lim 3x2  2 
(b)
(c) lim  4 x3  x2  7 x  2 
(d)
x 4
x2
lim ( 8) 
x 0
lim
x 1
5  2x 
But what happens when direct substitution DOES NOT WORK!?!???
Ex 10:
x 2  16
(a) lim
x 4
x4
(b)
x4 1
lim
x 1 x  1
x 9
x 3
(d)
lim
(c) lim
x 9
x  25
x 5
x  25
Ex 11 – One Sided Limits Using Algebra:
(a) lim 4  2 x
(b)
(c) lim 3cos x
(d)
x2
x 
tan x  sin x
x 0
x
(e) lim
Limits at Infinity
lim 2 x3  5x
x1
lim
x 2
x2  x  2
x2  2x
Let’s revisit an earlier problem – the graph of the rational function f ( x ) 
1
. Let’s tweak
x
it a little bit, and then look at the concept of infinite limits.
f ( x) 
1
1
x3
lim f ( x) 
x 
lim f ( x) 
x 
lim f ( x) 
x 3
lim f ( x) 
x 3
lim f ( x) 
x 3
And finally…..
I know what you’re thinking. What would the AP Calc class have to deal with?
lim
x 
x3  2 x 2  3x  9

4 x3  6 x 2  2 x  9
Special Trig Limits …..
sin x
1
x 0
x
lim
1  cos x
cos x  1
OR lim
0
x 0
x 0
x
x
lim
Examples – Find the limit if it exists:
lim
x 0
sin x
5x
lim
tan x
x
x 0
lim
7 sin x  3cos x  3
4x
lim
sec   1
 sec 
lim
cos x tan x
x
lim
3(1  cos x)
x
lim
cos x
cot x
lim
1  tan x
sin x  cos x
x 0
x 0
x

2
 0
x 0
x

4
Continuity
In order for a function to be continuous at a point, c, the THREE conditions below must
exist:
I.
f (c ) must exist
II.
lim f ( x ) must exist
x c
III.
lim f ( x ) must equal f (c ) .
x c
Now, let’s prove whether the following functions are continuous or not!
Example 1
Determine whether f ( x) is continuous at x = 2.
 x2  2x

 x  2
f ( x)  2
x4

 x  1
x2
x2
x2
Example 2
Determine whether f ( x) is continuous at x = 3.
3x  4
f ( x)  
x  2
0 x3
x3
Example 3
Determine whether the functions below are continuous or discontinuous. If there exists
a discontinuity at any point, identify the discontinuity as removable or non-removable.
(a)
f ( x) 
x2  9
4 x  12
(b)
f ( x) 
x3
x  5x  6
2