2.2 Limits Involving Infinity

2.2 Limits Involving Infinity

1f x

x

1lim 0x x

As the denominator gets larger, the value of the fraction gets smaller.

There is a horizontal asymptote if:

limx

f x b

or limx

f x b

2lim

1x

x

x

Example 1:

2limx

x

x

This number becomes insignificant as .x

limx

x

x 1

There is a horizontal asymptote at 1.

sin xf x

x

Example 2:

sinlimx

x

x Find:

When we graph this function, the limit appears to be zero.1 sin 1x

so for :0x 1 sin 1x

x x x

1 sin 1lim lim limx x x

x

x x x

sin0 lim 0

x

x

x

by the sandwich theorem:

sinlim 0x

x

x

Example 3: 5 sinlimx

x x

x

Find:

5 sinlimx

x x

x x

sinlim 5 limx x

x

x

5 0

5

Infinite Limits:

1f x

x

0

1limx x

As the denominator approaches zero, the value of the fraction gets very large.

If the denominator is positive then the fraction is positive.

0

1limx x

If the denominator is negative then the fraction is negative.

vertical asymptote at x=0.

Example 4:

20

1limx x

20

1limx x

The denominator is positive in both cases, so the limit is the same.

20

1 limx x

Humm….

End Behavior Models:

End behavior models model the behavior of a function as x approaches infinity or negative infinity.

A function g is:

a right end behavior model for f if and only if

lim 1x

f x

g x

a left end behavior model for f if and only if

lim 1x

f x

g x

Test ofmodel

Our modelis correct.

xf x x e Example 5:

As , approaches zero.x xe(The x term dominates.)

g x x becomes a right-end behavior model.

limx

x

x e

x

lim1

x

x

e

x

1 0 1

xh x e becomes a left-end behavior model.

limx

xx

x e

e

lim 1xx

x

e 0 1 1

As , increases faster than x decreases,x xe

therefore is dominant.xe

Test ofmodel

Our modelis correct.

xf x x e Example 5:

g x x becomes a right-end behavior model.

xh x e becomes a left-end behavior model.

On your calculator, graph:

1

2

3

x

x

y x

y e

y x e

10 10x

1 9y

Use:

5 4 2

2

2 1

3 5 7

x x xf x

x x

Example 6:

Right-end behavior models give us:

5

2

2

3

x

x

32

3

x

dominant terms in numerator and denominator

f(x)limx

Example 7:

021

42

27

5

xx

x

27

25

34132

)(xx

xxxf

Right-end behavior models give us:

dominant terms in numerator and denominator

f(x)limx

Example 8:

21

42

5

5

xx

25

25

34132

)(xx

xxxf

Right-end behavior models give us:

dominant terms in numerator and denominator

f(x)limx

Often you can just “think through” limits.

1lim sinx x

0

0lim sinx

x

0

1lim sinx x

zsinlim0z

0

Definition of a Limit Let c and L be real numbers. The function f has limit L as x approaches c

(x≠c), if, given any positive Ɛ, there is a positive number ɗ such that for all x, if x is within ɗ units of c, then f(x) is within Ɛ units of L.

0 < |x - c|< ɗ such that |f(x) – L| < Ɛ

Then we write Lf(x)limcx

Shortened version: Lf(x)lim

cx

If and only if for any number Ɛ >0, there is a real number ɗ >0such that if x is within ɗ units of c (but x ≠ c), then f(x) is withinƐ units of L.

C=2

L=3

3+Ɛ

3-Ɛ

→ ← ɗ = 1/3 ɗ is as large as possible. The graph just fits within the horizontal lines.

2274

)(2

x

xxxf

1. Plot the graph of f(x). Use a friendly window that includes x = 2 as a grid point. Name the feature present at x = 2.2. From the graph, what is the limit of f(x) as x approaches 2.3. What happens if you substitute x = 2 into the function?4. Factor f(x). What is the value of f(2)?5. How close to 2 would you have to keep x in order for f(x) tobe between 8.9 and 9.1?6. How close to 2 would you have to keep x in order for f(x) tobe within 0.001 unit of the limit in part 2? Answer in the form“x must be within ____ units of 2”7. What are the values of L, c, Ɛ and ɗ?8. Explain how you could find a suitable ɗ no matter how small Ɛ is.9. What is the reason for the restriction “… but not equal to c” in the definition of a limit?