Limits at Infinity and Continuity
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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.
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.2. The degree of the denominator is higher than the
degree of the numerator.
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.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
Denominator has higher power
• We start the same way; divide the numerator and denominator by the highest power of x in the denominator.
• Find:
33
3
33
2
3
2
52
4
524lim
xxx
xx
xx
xxx
x
Denominator has higher power
• We start the same way; divide the numerator and denominator by the highest power of x in the denominator.
• Find:
020
52
14
52
4
524lim
3
2
33
3
33
2
3
2
x
xx
xxx
xx
xx
xxx
x
Denominator has higher power
• 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
• We start the same way; divide the numerator and denominator by the highest power of x in the denominator.
• Find:
xxx
x 31125lim
23
Numerator has higher power
• We start the same way; divide the numerator and denominator by the highest power of x in the denominator.
• Find:
xx
x
xxx
xx
xxx
x 31
125
31125lim
23
23
Numerator has higher power
• We start the same way; divide the numerator and denominator by the highest power of x in the denominator.
• Find:
325
31
125
31125lim
2
23
23
xx
xx
x
xxx
xx
xxx
x
Numerator has higher power
• 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
Limits involving radicals
• Find:
3
8653lim
xx
x
3
8653lim
xx
x
Limits involving radicals
• Find:
33
21
8653lim
xx
x
3
8653lim
xx
x
Limits involving radicals
• Find:
632lim
2
xx
x
Limits involving radicals
• 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
Limits involving radicals
• Find:
632lim
2
xx
x||2 xx
31
63
21
||6
||3
2
lim222
2
x
x
xxx
xxx
x
Limits involving radicals
• Find:632lim
2
xx
x
||2 xx
Limits involving radicals
• 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
Limits involving radicals
• Find: )5(lim 36 xxx
Limits involving radicals
• Find: )5(lim 36 xxx
3636
3636
5
5lim)5(
)5()5(limxxxx
xxxxxx
Limits involving radicals
• Find: )5(lim 36 xxx
3636
3636
5
5lim)5(
)5()5(limxxxx
xxxxxx
0101
0lim3
3
66
6
3
5
5
xx
xxx
x
x
Limits involving radicals
• Find: )5(lim 336 xxxx
Limits involving radicals
• Find: )5(lim 336 xxxx
336
3
336
336336
5
5lim)5(
)5()5(limxxx
x
xxx
xxxxxxxx
Limits involving radicals
• Find: )5(lim 336 xxxx
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
Limits of exponential and logarithmic functions
• Let’s look at ex.
x
xelim 0lim
x
xe
Limits of exponential and logarithmic functions
• 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
.
Therefore the line 1 is a vertical asymptote.
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
Therefore the line 1 is a vertical asymptote.
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
What is the vertical asymptote?
Example 2: Find all vertical, horizontal, and slant asymptotes:
Limits at InfinityLimits at Infinity
132)( 2
xxxf
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 is a vertical asymptote at x = 0There is a vertical asymptote at x = 0
What is the vertical asymptote?
Example 3: Find all vertical, horizontal, and slant asymptotes:
Limits at InfinityLimits at Infinity
xxxf 12)(
What is the horizontal asymptote?
There is a horizontal asymptote at y = 2There is a horizontal asymptote at y = 2
What is the slant asymptote?
There are no slant asymptotesThere are no slant asymptotes
There are vertical asymptotes at x = -½ and -1There are vertical asymptotes at x = -½ and -1
What are the vertical asymptote?
Example 4: Find all vertical, horizontal, and slant asymptotes:
Limits at Infinity
132474)( 2
2
xxxxxf
What is the horizontal asymptote?
There is a horizontal asymptote at y = 2There is a horizontal asymptote at y = 2
What is the slant asymptote?
There are no slant asymptotesThere are no slant asymptotes
)1)(12(474 2
xxxx
There are vertical asymptotes at x = -2 and 2There are vertical asymptotes at x = -2 and 2
What are the vertical asymptote?
Example 5: Find all vertical, horizontal, and slant asymptotes:
Limits at Infinity
4273)( 2
2
xxxf
What is the horizontal asymptote?
There is a horizontal asymptote at y = 3There is a horizontal asymptote at y = 3
What is the slant asymptote?
There are no slant asymptotesThere are no slant asymptotes
)2)(2()3)(3(3
xxxx
There is a vertical asymptotes at x = -3/2There is a vertical asymptotes at x = -3/2
What are the vertical asymptote?
Example 6: Find all vertical, horizontal, and slant asymptotes:
Limits at Infinity
125212)( 2
2
xxxxxf
What is the horizontal asymptote?
There is a horizontal asymptote at y = ½ There is a horizontal asymptote at y = ½
What is the slant asymptote?
There are no slant asymptotesThere are no slant asymptotes
)4)(32()3)(4(
xxxx
)32()3(
xx
There is a vertical asymptotes at x = 4There is a vertical asymptotes at x = 4
What are the vertical asymptote?
Example 7: Find all vertical, horizontal, and slant asymptotes:
Limits at Infinity
xxxxxf
4113)( 2
23
What is the horizontal asymptote?
There are no horizontal asymptotesThere are no horizontal asymptotes
What is the slant asymptote?
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
What are the vertical asymptote?
Example 8: Find all vertical, horizontal, and slant asymptotes:
Limits at Infinity
28)(
3
xxxf
What is the horizontal asymptote?
There are no horizontal asymptotesThere are no horizontal asymptotes
What is the slant asymptote?
There are no slant asymptotesThere are no slant 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
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