Section 1.5: Infinite Limits Essay

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SECTION 1.5: INFINITE LIMITS
(Sounds kind of OXYMORONIC doesn¶t it?)

Goals: The Student Will Be Able To:

Recognize and properly note when and how infinite limits exist Explain what it is to increase and decrease without bound Define an Infinite Limit with formal symbolic notation Define and identify vertical asymptotes Distinguish between positive and negative infinity Apply standard properties of infinite limits

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INFINITE LIMITS

Increases and Decreases Without Bound:
Let f be a function, which is defined at every real number in some open interval containing c. A function increases without bound: lim f ( x ) ! g lim f ( x) ! g If x p c or x p c

A function decreases without bound:

If

xpc

lim f ( x ) ! g

or

xpc

lim f ( x) ! g

2

Here is an example of a function, which has both positive and negative infinite limits.

f ( x) ! lim f ( x ) ! g x p 2

x2 1 x2 4 lim f ( x ) ! g x p 2

x p 2

lim f ( x ) ! g

lim f ( x ) ! g x p 2

Here is a Graphical/Numerical Example:

3 f ( x) ! x2 lim f ( x ) ! g xp2 lim f ( x ) ! g x p 2

x approaches 2 from the left

x approaches 2 from the right

x f(x)

1.9 -30

1.9999 -3000

2 ?

2.0001 3000

2.1 30

f(x)decreases without bound

f(x)increases without bound

3

EXAMPLE 1: Find the left, right and overall limit of each: 1 f ( x) ! a. x 1

lim x p1

1 ! g x 1

lim x p1

1 !g x 1

Since the limits from the left and right are not the same, lim x p1

1 ! d .n.e x 1

b. x p1

f ( x) ! 1

1 ( x 1) 2 x p1

lim

( x 1)

2

!g

lim

1 ( x 1)
2

!g

Since the limits from the left and right are the same, 1 lim !g 2 x p1 ( x 1)

4

c.

f ( x) !

1 x 1 x p1

x p1

lim

1 !g x 1

lim

1 ! g x 1

Since the limits from the left and right are not the same, lim x p1

1 ! d .n.e x 1

d. x p1

f ( x) ! 1

1 ( x 1) 2 1 ( x 1)
2

lim

( x 1)

2

! g lim

! g

x p1

Since the limits from the left and right are the same, 1 lim ! g 2 x p1 ( x 1)

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VERTICAL ASYMPTOTES Definition: If f(x) approaches infinity (or negative infinity) as x approaches c from the left or the right, then the line x = c is a vertical asymptote of the graph of f(x). Theorem: Vertical Asymptotes: Let f and g be continuous functions on an open interval containing c. If f(c) { 0 and g(c) = 0, then the function:

f ( x) h( x ) ! g ( x) has a Vertical Asymptote at x = c. EXAMPLE 2: Finding Vertical Asymptotes Determine all the Vertical Asymptotes of the graph of each function. 1 f ( x) ! x = -1 a. 2( x 1)

x2 1 x2 1 f ( x) ! 2 ! b. x 1 ( x 1)( x 1) cos x f ( x ) ! cot x ! c. sin x

x = -1 & x = 1 x = nT

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EXAMPLE 3: Finding Vertical Asymptotes ± When there is a common factor. Determine all the Vertical Asymptotes of the graph of each function. x 2 2 x 8 ( x 4)( x 2) ( x 4) f ( x) ! ! ! 2 x 4 ( x 2)( x 2) ( x 2) So, in this case, even though x = 2 does make the denominator zero, it also makes the numerator zero as well. Hence it is not a V.A. In fact it IS a hole! Thus, according to our

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