Section 1.2 - Finding Limits Graphically and Numerically

LimitInformal Definition: If f(x) becomes arbitrarily close

to a single REAL number L as x approaches c from either side, the limit of f(x), as x appraches c, is L.

limx cf x L

The limit of f(x)…

as x approaches c…

is L.

Notation:

c

L

f(x)

x

Calculating Limits

Our book focuses on three ways:

1. Numerical Approach – Construct a table of values

2. Graphical Approach – Draw a graph

3. Analytic Approach – Use Algebra or calculus

This Lesson

Next Lesson

Example 1Use the graph and complete the table to find the limit (if it exists).

3

2limxx

3

2limxx

x 1.9 1.99 1.999 2 2.001 2.01 2.1

f(x) 6.859 7.88 7.988 8 9.2618.128.012

If the function is continuous at the value of x, the limit is easy to calculate.

8

Example 2Use the graph and complete the table to find the limit (if it exists).

2 111

lim xxx

2 111

lim xxx

x -1.1 -1.01 -1.001 -1 -.999 -.99 -.9

f(x) -2.1 -2.01 -2.001 DNE -1.9-1.99-1.999

If the function is not continuous at the value of x, a graph and table can be very useful.

2

Can’t divide by 0

-6

Example 3Use the graph and complete the table to find the limit (if it exists).

4

7 if 4

lim if 8 if 4

1 if 4x

x x

f x f x x

x x

4

limx

f x

x -4.1 -4.01 -4.001 -4 -3.999 -3.99 -3.9

f(x) 2.9 2.99 2.999 8 2.92.992.999

If the function is not continuous at the value of x, the important thing is what the output gets

closer to as x approaches the value.

3The limit does not change if the

value at -4 changes.

-6

Three Limits that Fail to Exist

f(x) approaches a different number from the right side of c than it approaches from the left side.

4

lim Does Not Existx

f x

Three Limits that Fail to Exist

f(x) increases or decreases without bound as x approaches c.

0

lim Does Not Existxf x

Three Limits that Fail to Exist

f(x) oscillates between two fixed values as x approaches c.

1

0limsin Does Not Existxx

x 0f(x) -1 1 -1 DNE 1 -1 1

2 2

3 25 2

52

32

Close

CloserClosest

A Limit that DOES ExistIf the domain is restricted (not infinite), the

limit of f(x) exists as x approaches an endpoint of the domain.

5

lim 5x

f x

Example 1Given the function t defined by the graph, find the limits at right.

4

3

0

6

2

5

1. lim

2. lim

3. lim

4. lim

5. lim

6. lim

x

x

x

x

x

x

t x

t x

t x

t x

t x

t x

2

3

DNE

3

DNE

2

t x

Example 2

Sketch a graph of the function with the following characteristics:

1. does not exist, Domain: [-2,3),

and Range: (1,5)

2. does not exist,

Domain: (-∞,-4)U(-4,∞), and

Range: (-∞,∞)

limx 0

f x

limx 0

f x

ClassworkSketch a graph and complete the table to find the limit (if it exists).

1

0lim 1

x

xx

1

0lim 1

x

xx

x -0.1 -0.01 -0.001 0 0.001 0.01 0.1

f(x) 2.8680 2.732 2.7196 DNE 2.59372.70482.7169

This a very important value that we will investigate more in Chapter 5. It deals with

natural logs.

e

Why is there a lot of “noise” over here?