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1.5 INFINITE LIMITSMs. Clark

AP CALCULUS

DATE:

INFINITE LIMITS

Graph the function given by 𝑓 𝑥 =3

𝑥−2

INFINITE LIMITS

This behavior is denoted as

lim𝑥→2−

3

𝑥−2=

lim𝑥→2+

3

𝑥−2=

A limit in which 𝑓(𝑥) increases or decreases without bound as 𝑥 approaches 𝑐 is called an infinite limit.

EXAMPLES

Determine the limit of each function as x approaches 1 from the left and from the right.

𝑓 𝑥 =1

𝑥−1 2 𝑓 𝑥 = −1

𝑥−1

VERTICAL ASYMPTOTESIf it were possible to extend the graphs in Figure 1.41 toward positive and negative infinity, you would see that each graph becomes arbitrarily close to the vertical line 𝑥 = 1. This is a vertical asymptote of the graph of 𝑓.

EXAMPLESDetermine all vertical asymptotes of the graph of each function.

A.) 𝑓 𝑥 =1

2(𝑥+1)B.) 𝑓 𝑥 =

𝑥2+1

𝑥2−1

C.) 𝑓 𝑥 = cot 𝑥 D.) 𝑓 𝑥 =𝑥+4

𝑥2+2𝑥−8

Find the limit.

A.) lim𝑥→5

𝑥+1

𝑥−5B.) lim

𝑥→3

𝑥3

𝑥3−27

C.) lim𝑥→−1+

𝑥2+7𝑥+6

𝑥2−4𝑥−5

D.) lim𝑥→5−

𝑥2+7𝑥+6

𝑥2−4𝑥−5

E.) lim𝑥→1+

𝑥2+7𝑥+6

𝑥2−4𝑥−5

PROPERTIES OF INFINITE LIMITS

MORE EXAMPLESA.) Because lim

𝑥→01 = and lim

𝑥→01/𝑥2 = ,

you can write

lim𝑥→0

1 +1

𝑥2= (Property 1, Thm 1.15)

B.) Because lim𝑥→1−

𝑥2 + 1 = and lim𝑥→1−

(cot 𝜋𝑥) =

You can write

lim𝑥→1−

𝑥2+1

cot 𝜋𝑥= (Property 3, Thm 1.15)

C.) Because lim𝑥→0+

3 = and lim𝑥→0+

cot 𝑥 =

You can write,

lim𝑥→0+

3 cot 𝑥 = (Property 2, Thm 1.15)

INTERMEDIATE VALUE THEOREM

Theorem 1.3 is an important theorem concerning the behavior of functions that are continuous on a closed interval.

The Intermediate Value Theorem states that for a continuous function f, if x takes on all values between a and b, f(x) must take on all values between f(a) and f(b).

As an example of the application of the Intermediate Value

Theorem, consider a person’s height. A girl is 5 feet tall on her

thirteenth birthday and 5 feet 7 inches tall on her fourteenth

birthday.

Then, for any height h between 5 feet and 5 feet 7 inches,

there must have been a time t when her height was exactly h.

This seems reasonable because human growth is continuous and

a person’s height does not abruptly change from one value to

another.

The Intermediate Value Theorem guarantees the existence of at

least one number c in the closed interval [a, b] .

There may, of course, be more than

one number c such that f(c) = k,

as shown in Figure 1.35.

Why does the function have to be continuous for the Intermediate Value Theorem to apply?

A function that is not continuous does not necessarily exhibit

the intermediate value property.

For example, the graph of the

function shown in Figure 1.36 jumps

over the horizontal line given by y = k,

and for this function there is no value

of c in [a, b] such that f(c) = k.

WHEN WILL THE IVT BE USEFUL?

The Intermediate Value Theorem often can be used to

locate the zeros of a function that is continuous on a closed

interval.

Specifically, if f is continuous on [a, b] and f(a) and f(b)

differ in sign, the Intermediate Value Theorem guarantees

the existence of at least one zero of f in the closed

interval [a, b] .

EXAMPLEUse the Intermediate Value Theorem to explain why the polynomial function 𝑓 𝑥 = 𝑥3 + 2𝑥 − 1 has a zero in the interval [0,1].

EXAMPLE

Verify that the IVT applies on the interval [5/2, 4] and find the value of

𝑐 guaranteed by the theorem for the function 𝑓 𝑥 =𝑥2+𝑥

𝑥−1, 𝑓 𝑐 = 6