1.2
Finding Limits Graphically and Numerically
An Introduction to Limits
Sketch the graph of the function.
x 8
f x  
x2
3
1.
x
y
 3 19
2 
1 7
0 4
1 3
2 4
3 7
4 12
5 19
x
y
 2.515.25
 2.313.89
 2.2 13.24
 2.112.61
 1.5 9.25
 1.7 10.29
 1.8 10.84
 1.9 11.41
 1.95 11.7
lim f  x   12
x  2
1.2
Finding Limits Graphically and Numerically
An Introduction to Limits
Definition of a limit:
We say that the limit of f(x) is L as x approaches a and write this
as
lim f x   L
xa
provided we can make f(x) as close to L as we want for all x
sufficiently close to a, from both sides, without actually
letting x be a.
Finding Limits Graphically and Numerically
1.2
Example 1 Estimating a limit numerically
Estimate the value of the following limit.
x  4 x  12
lim
2
x 2
x  2x
2
4
Limits are asking what the
function is doing around
x = a, and are not concerned
with what the function is
actually doing at x = a.
x
y
2.5 3.4
2.1 3.857
2.01 3.985
2.001 3.998
1.5 5.0
1.9 4.158
1.99 4.015
1.999 4.002
1.2
Finding Limits Graphically and Numerically
Example 2 Finding a limit
Estimate the value of the following limit.
lim g  x 
x2
4
 x 2  4 x  12

2
g x    x  2 x
6

, if x  2
, if x  2
1.2
Finding Limits Graphically and Numerically
Example 3 Behavior that differs from the right and left
Estimate the value of the following limit.
lim H t 
t 0
0 , if t  0
H t   
1 , if t  0
limit does not exist
1.2
Finding Limits Graphically and Numerically
12
90
8
7
-1
-2
-6
-7
-8
-9
6
5
4
3
2
1
0
-3
-4
-5
11
Example 4 Unbounded behavior
Estimate the value of the following limit.
3
lim
x  2 x  2
limit does not exist
1.2
Finding Limits Graphically and Numerically
12
90
8
7
-6
-7
-8
-9
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
11
Example 5 Oscillating behavior
Estimate the value of the following limit.
 
lim cos 
t 0
t 
limit does not exist
t
 1 
f
  1
1
 2001 
0.1
 2 
f
0
0
.
01
 2001 
0.001
f (t)
1
1
1
1
1 1
 .1 1
 .01 1
 .001
1
Finding Limits Graphically and Numerically
1.2
The Formal Definition of a Limit
Let f(x) be a function defined on an interval that contains x = a,
except possibly at x = a. Then we say that,
lim f x   L
xa
if for every number e > 0 there is some number d > 0 such that
whenever
f x L e
0 xa d

1.2
Finding Limits Graphically and Numerically
12
90
8
7
-6
-7
-8
-9
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
11
Example 6 Finding a d for a given e
Given the limit
lim 5 x  4   6
x2
find d such that
whenever
5x  4  6
0  x  2  d.
 0.01
1.2
Finding Limits Graphically and Numerically
1.1
A Preview of Calculus
Pg. 46, 1.1 #1-11