1.2 FINDING LIMITS

Numerically and Graphically

Limits• A function f(x) has a limit L as x approaches c if we can get

f(x) as close to c as possible but not equal to c.

x is very close to, not necessarily at, a certain number c

NOTATION:

limx c

f (x)

3 Ways to find Limits

• Numerically - construct a table of values and move arbitrarily close to c

• Graphically - exam the behavior of graph close to the c

• Analytically

1) Given , find

x 1.9 1.99 1.999 1.9999

f (x)

x 2.0001 2.001 2.01 2.1

f (x)

2

24

4

4

3.61 3.9601 3.996001 3.99960001

4.00040001 4.004001 4.0401 4.41

f (x) x 2

limx 2x 2

2) Given , find 1

1)(

3

x

xxf )(lim

1xf

x

x 0.9 0.99 0.999 0.9999

f (x)

x 1.0001 1.001 1.01 1.1

f (x)

1

13

3

3

2.710 2.9701 2.997001 2.99970001

3.00030001 3.003001 3.0301 3.31

3. What does the following table suggest about

a)

b)

)(1

limxf

x

)(1

limxf

x

x 0.9 0.99 0.999 1.001 1.01 1.1

F(x) 7 25 4317 3.0001 3.0047 3.01

Finding Limits Graphically• There is a hole in the graph.

Limits that Exist even though the function fails to Exist

One sided Limits

notation

1.Limits from the right

1.Limits from the left

)(lim

xfcx

)(lim

xfcx

4) Use the graph of to find

3

f ( x ) x 2 2

limx 1

( x 2 2)

5) Use the graph of to find

21

1)(

2

x

xxf

1

1lim

2

1

x

xx

0 1

0 1)(

x

xxf

)(lim0

xfx

limx 0

f (x)

6) Use the graph of to find

1

–1

1–1

Does Not Exist – DNE

limx 0

f (x)

limx 0

f (x)

Limits that Fail to Exist

• In order for a limit to exist the limit must be the same from both the left and right sides.

1

–1

1–1

Limits that Fail to Exist

• The behavior is unbounded or approaches an asymptote

1

–1

1–1

Limits that Fail to Exist

• The behavior oscillates

xx

1sin

0

lim

HOMEWORK

Page 54

# 1-10 all numerically

# 11 – 26 all graphically