1.2 Finding Limits Graphically
and Numerically
UNIT 1 DAY 2
Do Now
 Take a calculator.
 Fill in the table of values below for
x3 -1
f (x) =
, x ¹1.
x -1
x
f(x)
0.9 0.99 0.999
1

1.001 1.01
1.1
Graph
x3 -1
f (x) =
, x ¹1.
x -1
Introduction to Limits
 Consider the function


x3 -1
f (x) =
, x ¹1.
x -1
What happens to its graph at x = 1?
What happens to its graph near x = 1?
x approaches 1 from the right
x approaches 1 from the
left
x
0.9
0.99 0.999
f(x) 2.710 2.970
2.997
1
1.001
1.01
1.1

3.003
3.030 3.310
Informal Description of Limits
x3 -1
f (x) =
, x ¹1.
x -1
 Although x cannot equal 1, you can get as close to 1 as
you want.
 As we do so, f(x) gets closer and closer to ______.

Limit notation:
 Informal description of limit: If f(x) becomes
arbitrarily close to a single number, L, as x
approaches a given value, c, from either side, then the
limit of f(x), as x approaches c, is L.

Limit notation:
Example 1: Estimating a Limit Numerically
 Estimate
lim
x®0
x
x +1 -1
by evaluating the function at several points near x = 0.
x
f(x)
-0.01
-0.001
-0.0001
0

0.0001
0.001
0.01
Graph of Ex. 1
x
f (x) =
x +1 -1
Note
 In that last example, f(x) was undefined (does not
exist) at 0… but the limit of f(x) does exist at 0.
 The existence of f(x) at x = c has no bearing on the
existence of the limit of f(x) as x approaches c!
Example 2: Piecewise function
 Find the limit of f(x) as x approaches 2, where f is
defined as
ìï 1, x ¹ 2
f (x) = í
.
ïî 0, x = 2
Approaches for Finding Limit
 Numerical (table of values)
 Graphical
 Analytic (algebra/ calculus)
When Limits Fail to Exist
 When does a limit exist?
Ex. 3: Different Behavior from Left and Right
 Show that the limit does not exist.
x
lim
x®0 x
 If
lim- f (x) ¹ lim+ f (x), then the limit does not exist.
x®c
x®c
Ex. 4: Unbounded Behavior
 Discuss the existence of the limit
1
.
lim
2
x®0 x
Unit Circle Review
Ex. 5: Oscillating Behavior
 Discuss the existence of the limit.
1
lim sin
x®0
x
x
f(x)
2/
π
2/
3π
2/
5π
2/
7π
2/
9π
2/
11π
Unusual Limit Behavior
 The Dirichlet function:
ìï 0, if x is rational
f (x) = í
ïî 1, if x is irrational
 No limit at any real number c; therefore not
continuous at any real number c.
Closure
List three different types of behavior in which the
limit does not exist.
1.
lim- f (x) ¹ lim+ f (x)
x®c
x®c
2.
f(x) increases or decreases without bound as x  c
3.
f(x) oscillates between two fixed values as x  c