Finding Limits Graphically and Numerically

2015 Limits Introduction

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1.2

Estimate a limit using a numerical or graphical approach.

Learn different ways that a limit can fail to exist.

Study and use the informal definition of limit.

Objectives

Formal definition of a Limit:

If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c, is L.

“The limit of f of x as x approaches c is L.”

limx cf x L

This limit is written as

Limits can be found in various ways:

a) Graphically

b) Numerically

c) Algebraically

Ex: Find the following limit:

2

2lim 1xx

5

0

sinlimx

x

x

An Introduction to Limits

Ex: Find the following limit:

sin /y x x

Looks like y=1

0

sinlim 1x

x

x

3

1

1lim

1x

x

x

An Introduction to Limits

Ex: Find the following limit:

Start by sketching a graph of the function

For all values other than x = 1, you can use standard

curve-sketching techniques.

However, at x = 1, it is not clear what to expect.

We can find this limit numerically:

An Introduction to Limits

To get an idea of the behavior of the graph of f near x = 1, you can use two sets of x-values–one set that approaches 1 from the left and one set that approaches 1 from the right, as shown in the table.

An Introduction to Limits

The graph of f is a parabola that has a gap at the point (1, 3), as shown in the Figure 1.5.

Although x can not equal 1, you can move arbitrarily close to 1, and as a result f(x) moves arbitrarily close to 3.

Using limit notation, you can write

An Introduction to Limits

This is read as “the limit of f(x) as x approaches 1 is 3.”

Figure 1.5

This discussion leads to an informal definition of a limit:

A limit is the value (meaning y value) a function approaches as x approaches a particular value from the left and from the right.

Properties of Limits:

Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power.

For a limit to exist, the function must approach the same value from both sides.

One-sided limits approach from either the left or right side only.

x clim f x L

x clim f x L

x clim f x L

The limit of a function refers to the value that the function approaches, not the actual value (if any).

2

limx

f x

not 1

2

y f x

1 2 3 4

1

2

At x=1: 1

lim 0x

f x

1

lim 1x

f x

1 1f

left hand limit

right hand limit

value of the function

1

limxf x

does not exist because the left and right hand limits do not match!

1limxf x

DNE

y f x

At x=2: 2

lim 1x

f x

2

lim 1x

f x

2 2f

left hand limit

right hand limit

value of the function

2

lim 1x

f x

because the left and right hand limits match.

1 2 3 4

1

2

2

lim 1xf x

y f x

At x=3: 3

lim 2x

f x

3

lim 2x

f x

3 2f

left hand limit

right hand limit

value of the function

3

lim 2xf x

because the left and right hand limits match.

1 2 3 4

1

2

3

lim 2xf x

y f x

You Try– Estimating a Limit Numerically

Use the table feature of your graphing calculator to

evaluate the function at several

points near x = 0 and use the results to estimate the limit:

Example 1 – Solution

The table lists the values of f(x) for several x-values near 0.

0

lim 2xf x

Example 1 – Solution

From the results shown in the table, you can estimate the limit to be 2.

This limit is reinforced by the graph of f (see Figure 1.6.)

cont’d

Figure 1.6

Use a graphing utility to estimate the limit:

3 2

1

1lim

1x

x x x

x

Find

2

1

2

limx

f x

1

1, 20, 2xxf x

2

limx

f x

Limits That Fail to Exist

Show that does not exist

1

1

Non-existance

x

f xx

0limx

x

x

Because the behavior differs from the right and from the left of zero, the limit DNE.

Discuss the existence of the limit:

Solution: Using a graphical representation, you can see that x does not approach any number. Therefore, the limit does not exist.

0limx

20

1limx x

0limx

0limxDNE

Fig. 1.10, p. 51

0limxDNE

Limits That Fail to Exist - 3 Reasons

Properties of Limits

Limits Basics

limx cx c

limx cb b

lim n n

x cx c

lim n n

x cx c

3lim 2x

2

5limxx

5

2

3limxx

9

3

8limxx

2

Examples

Properties of Limits

Using Properties of Limits

3

limx

f x g x

1

2

3limx

g x

3

limx

f x g x

5

84

2 3

3 3

lim 7 lim 12x x

If f x and g x

• Find the following limits:

2

4limxx

4lim5x

x

9limx

x

lim cosxx x

16

3

20

2

1lim 6x

x x

6

2

5

3 10lim

5x

x x

x

7

Properties of Limits

Compute the following limits

2

1lim

1x x

3

3lim

1x

x

x

sin 2limx

x

x

21

1lim

2x

x

x x

4

lim tanx

x

lim cosxx x

• Let’s take a look at the last one

• What happened when we plugged in 1 for x?

• When we get we have what’s called an

indeterminate form

• Let’s see how we can solve it

2

1lim

21

xx

xx

0

0

• Let’s look at the graph of

Is the function continuous at x = 1?

2

1lim

21

xx

xx

You Try:

Find the limit:

5

You Try:

3

3

27Evaluate lim

3x

x

x

27

Find the limit:

Solution:

By direct substitution, you obtain the indeterminate form 0/0.

Example – Rationalizing Technique

In this case, you can rewrite the fraction by rationalizing the numerator.

cont’dSolution

Now, using Theorem 1.7, you can evaluate the limit as shown.

cont’dSolution

A table or a graph can reinforce your conclusion that the

limit is . (See Figure 1.20.)

Figure 1.20

Solutioncont’d

Solutioncont’d

Group Work:Sketch the graph of f. Identify the values of c for which exists. lim

x cf x

lim exists for all values except where 4.x cf x c

Homework

p.54 8-25 all