MAT 1234Calculus I

Section 3.4

Limit at infinity

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Preview

We want to examine the behavior of a function when is getting bigger and bigger , i.e. as

We will look at how to evaluate these kind of limits

We will use an intuitive approach (skip the precise definition at the end of the section)

Preview

We want to examine the behavior of a function when is getting bigger and bigger , i.e. as

We will look at how to evaluate these kind of limits

We will use an intuitive approach (skip the precise definition at the end of the section)

Caution on infinity

Infinity, , is NOT a number It does not make sense to do, e.g.

It is a concept

Example 12)( xxfy

2limx

x

2lim

xx

Example 23)( xxfy

3limxx

3limx

x

Example 3 x

xxfy

1)(

1limx

x

x

1limx

x

x

horizontal asymptote

Remarks

In the example above, the graph suggests what the limit should be.

We will use the following theorem to compute limits. (Graphs are used as confirmations.)

Theorem

If is a rational no., then

(a)

(b) If is defined for all x, then

1lim 0

rx x

1lim 0

rx x

Example 4

2

1)(x

xfy

2

1limx x

2

1limx x

1 1lim 0, lim 0

r rx xx x

Example 5

xxxfy

11)(

2/1

1/2 1/2

1 1lim , limx xx x

1 1lim 0, lim 0

r rx xx x

Example 6

1

1lim

2

2

x

xx

Find the limit

Example 6 1

1lim

2

2

x

xx

In order to take advantage of the theorem, we need to rewrite the function so that we have terms in the form of

where is a positive rational no.

rx

1

1 1lim 0, lim 0

r rx xx x

Example 6

2

2

1lim

1x

x

x

To do that, we divide both the numerator and the denominator by the highest power of in the denominator.

rx

1

1

1lim

2

2

x

xx

1 1lim 0, lim 0

r rx xx x

Example 7 1

1lim

4

3

x

xxx

3

4

1lim

1x

x x

x

1 1lim 0, lim 0

r rx xx x

Example 8 2

32lim

2

x

xx

22 3lim

2x

x

x

1 1lim 0, lim 0

r rx xx x

Remark

We cannot use the limit laws. The resulting expression is “meaningless”. In this situation, we will look at the “behavior”

of the limit as .

23

22 3lim lim

0

02 121

x x

xx xx

x

Remark

Note that we did not and cannot use the limit laws in this example.

We used an “educational guess” which is acceptable for these type of problems at this level.

If you want to see a better solution, which are not required for this class, it is in the next slide (hidden).

Example 9 2

1lim

2

x

xx

Example 9 2

1lim

2

x

xx

22

2 2

2

111

lim lim lim2 22

11

1 0lim 1

2 1 01

x x x

x

xxx xx

x xxx x

x

x

Example 9 2

1lim

2

x

xx

12

1lim

2

x

xx

Why?

Example 9 Correct Version

22

2 2

2

111

lim lim lim2 22

11

1 0lim 1

2 1 01

x x x

x

xxx xx

x xxx x

x

x