Limits Involving InfinityChapter 2: Limits and Continuity

What you’ll learn about

• Finite Limits as x→±∞• Sandwich Theorem Revisited• Infinite Limits as x→a• End Behavior Models

…and whyLimits can be used to describe the behavior of functionsfor numbers large in absolute value.

Finite limits as x→±∞

The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds.For example, when we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line.When we say “the limit of f as x approaches negative infinity (- ∞)” we mean the limit of f as x moves increasingly far to the left on the number line.

The line is a of the graph of a function

if either

lim or limx x

y b

y f x

f x b f x b

horizontal asymptote

Example 1 - Horizontal Asymptote

Use a graph and tables to find a lim and b lim .

c Identify all horizontal asymptotes.

1

x xf x f x

xf x

x

a lim 1

b lim 1

c Identify all horizontal asymptotes. 1

x

x

f x

f x

y

By looking at the graph and a table of values, it appears that as we head off in both the positive and negative x directions towards infinity, that the graph gets closer and closer to 1. Thinking about the function f(x), it makes sense that the function will never actually equal 1 in either direction as we head to infinity but will just continue to get infinitely closer to 1. Using the graph, the table, and our intuition about the given function f(x), we can give the following answers:

Note that we couldn’t just use direct substitution here. Plugging in ∞ for x doesn’t make any sense.

Example 2 - Sandwich Theorem Revisited

The sandwich theorem also holds for limits as .

cosFind lim graphically and using a table of values.

x

x

x

x

The graph and table suggest that the function oscillates about the -axis.

cosThus 0 is the horizontal asymptote and lim 0

x

x

xy

x

First off, how are we going to use the Sandwich Theorem? Well, let’s think about for a second. The largest it ever becomes is 1 and the smallest is -1. So we could then say: for all x. Now think about and as x approaches infinity. As the denominators gets larger and larger, and get closer and closer to 0. The same thing happens as x approaches negative infinity. Since is sandwiched between these two functions, it must also approach 0 as x approaches ±∞.

A lot of students don’t like this problem because they learned that a function can never cross a horizontal asymptote. Just think of an asymptote as something that the function gets closer to as we head to ±∞, don’t worry about the function crossing it.

Properties of Limits as x→±∞

If , and are real numbers and

lim and lim , then

1. : lim

The limit of the sum of two functions is the sum of their limits.

2. : lim

The limi

x x

x

x

L M k

f x L g x M

Sum Rule f x g x L M

Difference Rule f x g x L M

t of the difference of two functions is the difference

of their limits

Properties of Limits as x→±∞

Constant Multiple Rule:

Product Rule:

Properties of Limits as x→±∞

6. : If and are integers, 0, then

lim

provided that is a real number.

The limit of a rational power of a function is that power of the

limit of the function, provided the latt

rrss

x

r

s

Power Rule r s s

f x L

L

er is a real number.

These properties are essentially the same (except they apply as x → ∞) as the original properties that we learned for limits as x → c. You will do examples in your homework that will require you to use these rules.

End Behavior Models

The function is

a a for if and only if lim 1.

b a for if and only if lim 1.

x

x

g

f xf

g x

f xf

g x

right end behavior model

left end behavior model

1

If one function provides both a left and right end behavior model, it is simply called

an .

In general, is an end behavior model for the polynomial function nn

n nn n

g x a x

f x a x a x

end behavior model

10... , 0

Overall, all polynomials behave like monomials.na a

This is a pretty technical definition and we won’t really use it in practice much but I will demonstrate it in an example in an upcoming slide. The purpose of an end behavior model is really just to help us visualize what is happening as the function heads off to ±∞.

As a quick example of the rule in the box above, if I asked you what looked like as it might be a little hard to visualize it. But if I said that its end behavior model was you could visualize what happens to as . As , shoots off up to in the y direction.

Example 3 - End Behavior Models

2

2

Find an end behavior model for

3 2 5

4 7

x xf x

x

2

2

2

2

Notice that 3 is an end behavior model for the numerator of , and

4 is one for the denominator. This makes

3 3= an end behavior model for .

4 4

x f

x

xf

x

3In this example, the end behavior model for , is also a horizontal

4asymptote of the graph of . We can use the end behavior model of a

rational function to identify any horizontal asymptote.

f y

f

A rational function always has a simple power function as

an end behavior model.

In general, when we are looking at end behavior models of a rational function, just circle the term with the highest power of x in the numerator and the denominator. Simplify those terms with respect to each other and that will be your end behavior model. This will also help find any horizontal asymptotes that may exist.

Example 4 - End Behavior ModelsFind a left and right end behavior model for

Without end models, it’s kind of hard to visualize this function as we head off to ±∞.

Let’s focus on the right end behavior model first. Ask yourself, “As what is happening to f(x)?” Well in this case, is getting really big and is getting extremely small. We could say that is dominating the function as is getting infinitely close to 0. So at extremely large values of x, the function is behaving almost exactly the same as . So our right end behavior model is .

Now let’s look for the left end behavior model.Ask yourself, “As what is happening to f(x)?” Now the exact opposite is happening, plugging in a “very large” negative value for x will make get infinitely close to 0 and will make get extremely large. As , the becomes irrelevant. Our left end behavior model is .

Note:=1+0 so

Infinite Limits as x→a

If the values of a function ( ) outgrow all positive bounds as approaches

a finite number , we say that lim . If the values of become large

and negative, exceeding all negative bounds as x a

f x x

a f x f

approaches a finite number ,

we say that lim . x a

x a

f x

The line is a of the graph of a function

if either

lim or lim x a x a

x a

y f x

f x f x

vertical asymptote

In the previous sections, we had said that when this occurred, that the limit D.N.E.(does not exist). We are now kind of replacing that when both the left and right limits approach the same ±∞ from both sides. For AP exam purposes, saying D.N.E or ±∞ are often interchangeable. An exception to this is when one of the one sided limits goes to positive ∞ and the other goes to negative ∞. In that case there, we would still need to say that that the limit D. N. E.

A vertical asymptote of a function will occur at any x value when direct substitution yields where c is a constant (c≠0).

Example 5 - Vertical Asymptote

2

Find the vertical asymptotes of the graph of ( ) and describe the behavior

of ( ) to the right and left of each vertical asymptote.

8

4

f x

f x

f xx

Let’s take a look at the graph of the function.

The values of the function approach to the left of 2.

The values of the function approach + to the right of 2.

The values of the function approach + to the left of 2.

The values of the funct

x

x

x

2 22 2

2 22 2

ion approach to the right of 2.

8 8lim and lim

4 48 8

lim and lim4 4

So, the vertical asymptotes are 2 and 2

x x

x x

x

x x

x xx x

Remember what I said on the last slide: horizontal asymptotes will occur where the function takes the form where c is a non-zero constant. In this case here, that occurs when x = ±2.

Summary• The line is a horizontal asymptote of the graph of the function if

or

• There are properties of limits as that are similar to the regular properties of limits that we learned about in previous lessons.

• End behavior models can be used to describe the behavior of a function as . • For rational functions, these models can also be used to find horizontal asymptotes if

they exist. • Vertical asymptotes can be found when

or • Another way of saying this is: a vertical asymptote of a function will occur at any x value

when direct substitution yields where c is a constant (c≠0).