Limits at Infinity and Continuity

Limits at InfinityLimits at Infinity“Explore the End Behavior

of a Function”

End BehaviorEnd Behavior

• Observe the graph of the Rational Function, 1/x

01lim xx

01lim xx

Another ExampleAnother Example

2tanlim 1

x

x 2tanlim 1

x

x

Limits of Polynomials at InfinityLimits of Polynomials at Infinity

• If the leading coefficient of the polynomial is positive:

n

xxlim

n

xxlim

,...5,3,1, n

,...6,4,2, n

Limits of Polynomials at InfinityLimits of Polynomials at Infinity• If the leading coefficient of the polynomial is

negative:

n

xxlim

n

xxlim ,...5,3,1, n

,...6,4,2, n

Limits of Polynomials at InfinityLimits of Polynomials at Infinity

• The end behavior of a polynomial matches the end behavior of its highest degree term.

• For example:

5lim xx

162lim 45 xxxx

Limits of Rational Functions Limits of Rational Functions at Infinityat Infinity

• We will have three different situations.1. The degree of the numerator and the denominator is

the same.



the same.2. The degree of the denominator is higher than the

degree of the numerator.



the same.2. The degree of the denominator is higher than the

degree of the numerator.3. The degree of the numerator is higher than the

degree of the denominator.

Digging deeper…Digging deeper…

• Infinity is a very special idea. We know we can't reach it, but we can still try to work out the value of functions that have infinity in them.

• Question: What is the value of 1/∞ ?• Answer: We don't know!• Maybe we could say that 1/∞ = 0, ...

but if we divide 1 into infinite pieces and they end up 0 each, what happened to the 1?

• In fact 1/∞ is known to be undefined.

But We Can Approach It!x 1/x

1 1.000002 0.500004 0.25000

10 0.10000100 0.01000

1,000 0.0010010,000 0.00010

• The limit of 1/x as x approaches Infinity is 0

• Furthermore:

01lim nx x

Limits at Infinity: DefinitionLimits at Infinity: Definition

For all n > 0,

1 1lim lim 0n nx xx x

provided that is defined.1nx

Same Degree

• Find:8653lim

xx

x

Same Degree

• Find:

• We will divide the numerator and denominator by the highest power of x in the denominator.

8653lim

xx

x

x

xx 86

53lim

Same Degree

• Find:

• We will divide the numerator and denominator by the highest power of x in the denominator.

8653lim

xx

x

21

63lim86

53lim

xx

x

x

Same Degree

• When the numerator and the denominator are the same degree, your answer is a number, and it is the quotient of the leading coefficients.

Denominator has higher power

• We start the same way; divide the numerator and denominator by the highest power of x in the denominator.

• Find:

524lim 3

2

xxx

x



• Find:

33

3

33

2

3

2

52

4

524lim

xxx

xx

xx

xxx

x



• Find:

020

52

14

52

4

524lim

3

2

33

3

33

2

3

2

x

xx

xxx

xx

xx

xxx

x


• Conclusion: When the denominator has the highest power of x, the answer is zero.

• For fractions, if the denominator accelerates faster than the numerator, the limit is zero.

Numerator has higher power


• Find:

xxx

x 31125lim

23



• Find:

xx

x

xxx

xx

xxx

x 31

125

31125lim

23

23



• Find:

325

31

125

31125lim

2

23

23

xx

xx

x

xxx

xx

xxx

x


• Conclusion: When the numerator is the higher power of x, the answer is .

• In the past example, the function reacts like a negative second degree equation, so the answer is

.

35

325lim

22 xxxx

Limits at InfinityLimits at InfinityExample

2

23 5 1lim

2 4x

x xx

2

2

5 13lim 2 4x

x x

x

3 0 0 30 4 4

Divide by 2x

2

2

5 1lim 3 lim lim

2lim lim 4

x x x

x x

x x

x

More Examples

3 2

3 2

2 3 21. lim100 1x

x xx x x

3 2

3 3 3

3 2

3 3 3 3

2 3 2

lim100 1x

x xx x x

x x xx x x x

3

2 3

3 22lim 1 100 11x

x x

x x x

2 21

0

2

3 2

4 5 212. lim7 5 10 1x

x xx x x

2

3 3 3

3 2

3 3 3 3

4 5 21

lim7 5 10 1x

x xx x x

x x xx x x x

2 3

2 3

4 5 21

lim 5 10 17x

x x x

x x x

07

2 2 43. lim12 31x

x xx

2 2 4

lim12 31x

x xx x xxx x

42lim 3112x

xx

x

212

24. lim 1x

x x

22

2

1 1 lim1 1x

x x x x

x x

2 2

2

1lim1x

x x

x x

2

1 lim1x x x

1 1 0

Limits involving radicals

• Find:3

8653lim

xx

x


• Find:

3

8653lim

xx

x

3

8653lim

xx

x


• Find:

33

21

8653lim

xx

x

3

8653lim

xx

x


• Find:

632lim

2

xx

x


• Find:

632lim

2

xx

x||2 xx

||6

||3

2

lim22

2

xxx

xxx

x

Limits involving radicals• Find:

• If x is approaching positive infinity, the absolute value of x is just x.

632lim

2

xx

x||2 xx

xxxx

xxx

xxx

x 63

21

||6

||3

2

lim222

2


• Find:

632lim

2

xx

x||2 xx

31

63

21

||6

||3

2

lim222

2

x

x

xxx

xxx

x


• Find:632lim

2

xx

x

||2 xx


• Find:632lim

2

xx

x

||6

||3

2

lim22

2

xxx

xxx

x

||2 xx

Limits involving radicals• Find:

• If x is approaching negative infinity, the absolute value of x is the opposite of x.

632lim

2

xx

x

31

63

21

||6

||3

2

lim222

2

xxx

x

xxx

xxx

x

||2 xx


• Find: )5(lim 36 xxx



3636

3636

5

5lim)5(

)5()5(limxxxx

xxxxxx



3636

3636

5

5lim)5(

)5()5(limxxxx

xxxxxx

0101

0lim3

3

66

6

3

5

5

xx

xxx

x

x


• Find: )5(lim 336 xxxx



336

3

336

336336

5

5lim)5(

)5()5(limxxx

x

xxx

xxxxxxxx



336

3

336

336336

5

5lim)5(

)5()5(limxxx

x

xxx

xxxxxxxx

25

1015lim

3

3

6

3

6

6

3

3

5

5

xx

xx

xx

xx

x

Limits of exponential and logarithmic functions

xx

lnlim x

xlnlim

0


• Let’s look at ex.

x

xelim 0lim

x

xe


• Let’s look at e-x

0lim

x

xe

x

xelim

Infinite Limits: One-sidedInfinite Limits: One-sidedFor all n > 0,

1lim nx a x a

1lim if is evennx a

nx a

1lim if is oddnx a

nx a

-8 -6 -4 -2 2

-20

-15

-10

-5

5

10

15

20

-2 2 4 6

-20

-10

10

20

30

40

-8 -6 -4 -2 2

-20

-15

-10

-5

5

10

15

20

Examples Find the limits

2

20

3 2 11. lim2x

x xx

2

0

2 13= lim

2x

x x

32

3

2 12. lim2 6x

xx

3

2 1= lim2( 3)x

xx

-8 -6 -4 -2 2

-20

20

40

AsymptotesAsymptoteshorizontal asymptotThe line is called a

of the curve ( ) if eitherey L

y f x

lim ( ) or lim ( ) .x x

f x L f x L

vertical asymptote The line is called a of the curve ( ) if either

x cy f x

lim ( ) or lim ( ) .x c x c

f x f x

ExamplesExamplesFind the asymptotes of the graphs of the functions

2

2

11. ( )1

xf xx

1 (i) lim ( )

xf x

Therefore the line 1 is a vertical asymptote.

x

1.(iii) lim ( )x

f x

1(ii) lim ( )

xf x

.


x

Therefore the line 1 is a horizonatl asymptote.

y

-4 -2 2 4

-10

-7.5

-5

-2.5

2.5

5

7.5

10

2

12. ( ) 1

xf xx

21 1

1(i) lim ( ) lim 1x x

xf xx

1 1

1 1 1= lim lim .( 1)( 1) 1 2x x

xx x x

Therefore the line 1 is a vertical asympNO t eT ot .

x

1(ii) lim ( ) .

xf x


x

(iii) lim ( ) 0.x

f x

Therefore the line 0 is a horizonatl asymptote.

y

-4 -2 2 4

-10

-7.5

-5

-2.5

2.5

5

7.5

10

Continuity A function f is continuous at the point x = a if the following are true:

) ( ) is definedi f a) lim ( ) existsx a

ii f x

a

f(a)

A function f is continuous at the point x = a if the following are true:

) ( ) is definedi f a) lim ( ) existsx a

ii f x

) lim ( ) ( )x a

iii f x f a

a

f(a)

At which value(s) of x is the given function discontinuous?

1. ( ) 2f x x 2

92. ( )3

xg xx

Continuous everywhere Continuous everywhere

except at 3x

( 3) is undefinedg

lim( 2) 2 x a

x a

and so lim ( ) ( )x a

f x f a

-4 -2 2 4

-2

2

4

6

-6 -4 -2 2 4

-10

-8

-6

-4

-2

2

4

Examples

2, if 13. ( )

1, if 1x x

h xx

1lim ( )xh x

and

Thus h is not cont. at x=1.

11

lim ( )xh x

3

h is continuous everywhere else

1, if 04. ( )

1, if 0x

F xx

0lim ( )x

F x

1 and

0lim ( )x

F x

1

Thus F is not cont. at 0.x

F is continuous everywhere else

-2 2 4

-3

-2

-1

1

2

3

4

5

-10 -5 5 10

-3

-2

-1

1

2

3

Continuous Functions

A polynomial function y = P(x) is continuous at every point x.

A rational function is continuous at every point x in its domain.

( )( ) ( )p xR x q x

If f and g are continuous at x = a, then

, , and ( ) 0 are continuous

at

ff g fg g agx a

Intermediate Value Theorem

If f is a continuous function on a closed interval [a, b] and L is any number between f (a) and f (b), then there is at least one number c in [a, b] such that f(c) = L.

( )y f x

a b

f (a)

f (b)L

c

f (c) =

Example

2Given ( ) 3 2 5,Show that ( ) 0 has a solution on 1,2 .

f x x xf x

(1) 4 0(2) 3 0ff

f (x) is continuous (polynomial) and since f (1) < 0 and f (2) > 0, by the Intermediate Value Theorem there exists a c on [1, 2] such that f (c) = 0.

62

Limits and Horizontal Limits and Horizontal Asymptotes:Asymptotes:

Limits at Infinity RevisitedLimits at Infinity Revisited

Limits at InfinityLimits at Infinity• This is a topic that we already covered, but This is a topic that we already covered, but

we will look at it from a different perspective.we will look at it from a different perspective.• When we determine limits at infinity, we are

generally trying to find a functions “end end behaviorbehavior”

• When we examine a function’s end behavior, we want to know if it has horizontal or slant horizontal or slant asymptotesasymptotes.

• Recall the following rules for a function of the form:

011

1

011

1

......)(

bxbxbxbaxaxaxaxf m

mm

m

nn

nn

Limits at InfinityLimits at Infinity If you examine the degree of the numerator and denominator, there’s one of

three possibilities:

1. n > m: The degree of the numerator is bigger than the denominator (top heavy)a)a) LL =

2. n < m: The degree of the numerator is smaller than the denominator (bottom heavy)a)a) LL = 0

3. n = m: The degree of the numerator is equal to the denominator

a)a) L L =4. n = m + 1: The degree of the numerator is exactly one

more than the denominatora)a) L L = the quotient using long division

m

n

ba

The horizontal asymptote is The horizontal asymptote is yy = 0 = 0

The horizontal asymptote is The horizontal asymptote is yy = =

There is no horizontal asymptoteThere is no horizontal asymptote

m

n

ba

Not horizontal, but Not horizontal, but a slant asymptotea slant asymptote

There is a vertical asymptote at x = 0There is a vertical asymptote at x = 0

What is the vertical asymptote?

Example 1: Find all vertical, horizontal, and slant asymptotes:

Limits at InfinityLimits at Infinity

xxf 1)(

What is the horizontal asymptote?

There is a horizontal asymptote at y = 0There is a horizontal asymptote at y = 0

What is the slant asymptote?

There are no slant asymptotesThere are no slant asymptotes

There are no vertical asymptotesThere are no vertical asymptotes




132)( 2

xxxf





There is a vertical asymptote at x = 0There is a vertical asymptote at x = 0




xxxf 12)(





There are vertical asymptotes at x = -½ and -1There are vertical asymptotes at x = -½ and -1

What are the vertical asymptote?


Limits at Infinity

132474)( 2

2

xxxxxf





)1)(12(474 2

xxxx

There are vertical asymptotes at x = -2 and 2There are vertical asymptotes at x = -2 and 2



Limits at Infinity

4273)( 2

2

xxxf





)2)(2()3)(3(3

xxxx

There is a vertical asymptotes at x = -3/2There is a vertical asymptotes at x = -3/2



Limits at Infinity

125212)( 2

2

xxxxxf


There is a horizontal asymptote at y = ½ There is a horizontal asymptote at y = ½



)4)(32()3)(4(

xxxx

)32()3(

xx

There is a vertical asymptotes at x = 4There is a vertical asymptotes at x = 4



Limits at Infinity

xxxxxf

4113)( 2

23


There are no horizontal asymptotesThere are no horizontal asymptotes


There is a slant asymptote at y = 3x + 1 by long divisionThere is a slant asymptote at y = 3x + 1 by long division

)4()113(2

xxxx

4)113(

xxx

There are no vertical asymptotesThere are no vertical asymptotes



Limits at Infinity

28)(

3

xxxf


There are no horizontal asymptotesThere are no horizontal asymptotes



2)42)(2( 2

x

xxx422 xx

2

2 3

3lim

xxx

xx

xx

x

xxx

xx

xx

x 3

3lim

2

xxx

xxxx 3

3lim2

22

Example 9: Here’s a tough one – Evaluate:

Limits at Infinity

xxxx

3lim 2

xxx

xxx

3

32

2

xxx

xxxx 3

3lim2

222

xxx

xx 3

3lim2

2xx thus,

0;x ,x since :Note

22

2 31

3lim

xx

xxx

Divide everything by xDivide everything by x

0113

x

x 311

3lim

23

Apply the limitApply the limit

Limits at Infinity and Continuity

Documents

higher power of x

highest power of x

higher powerwe

highest degree term

higher powerconclusion

x approaches infinity

limits of rational functions

different situations