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We examine two ways of extending the definition of limit: A function can be said to have a limit of infinity (or minus infinity) at a point if it grows without bound near that point. A function can have a limit at a point if values of the function get close to a value as the points get arbitrarily large.
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Section 2.5Limits Involving Infinity
Math 1a
February 4, 2008
Announcements
I Syllabus available on course website
I All HW on website now
I No class Monday 2/18
I ALEKS due Wednesday 2/20
Outline
Infinite LimitsVertical AsymptotesInfinite Limits we KnowLimit “Laws” with Infinite LimitsIndeterminate Limits
Limits at InfinityAlgebraic rates of growthExponential rates of growthRationalizing to get a limit
Worksheet
Infinite Limits
DefinitionThe notation
limx→a
f (x) =∞
means that the values of f (x) can be made arbitrarily large (aslarge as we please) by taking x sufficiently close to a but not equalto a.
DefinitionThe notation
limx→a
f (x) = −∞
means that the values of f (x) can be made arbitrarily largenegative by taking x sufficiently close to a but not equal to a.
Of course we have definitions for left- and right-hand infinite limits.
Vertical Asymptotes
DefinitionThe line x = a is called a vertical asymptote of the curvey = f (x) if at least one of the following is true:
I limx→a
f (x) =∞I lim
x→a+f (x) =∞
I limx→a−
f (x) =∞
I limx→a
f (x) = −∞I lim
x→a+f (x) = −∞
I limx→a−
f (x) = −∞
Infinite Limits we Know
limx→0+
1
x=∞
limx→0−
1
x= −∞
limx→0
1
x2=∞
Finding limits at trouble spots
Example
Let
f (t) =t2 + 2
t2 − 3t + 2
Find limt→a−
f (t) and limt→a+
f (t) for each a at which f is not
continuous.
SolutionThe denominator factors as (t − 1)(t − 2). We can record thesigns of the factors on the number line.
Finding limits at trouble spots
Example
Let
f (t) =t2 + 2
t2 − 3t + 2
Find limt→a−
f (t) and limt→a+
f (t) for each a at which f is not
continuous.
SolutionThe denominator factors as (t − 1)(t − 2). We can record thesigns of the factors on the number line.
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ − ∓∞ +
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ − ∓∞ +
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ − ∓∞ +
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ − ∓∞ +
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+
±∞ − ∓∞ +
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞
− ∓∞ +
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ −
∓∞ +
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ − ∓∞
+
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ − ∓∞ +
Limit Laws with infinite limitsTo aid your intuition
I The sum of positive infinite limits is ∞. That is
∞+∞ =∞
I The sum of negative infinite limits is −∞.
−∞−∞ = −∞
I The sum of a finite limit and an infinite limit is infinite.
a +∞ =∞a−∞ = −∞
Rules of Thumb with infinite limitsDon’t try this at home!
I The sum of positive infinite limits is ∞. That is
∞+∞ =∞
I The sum of negative infinite limits is −∞.
−∞−∞ = −∞
I The sum of a finite limit and an infinite limit is infinite.
a +∞ =∞a−∞ = −∞
Rules of Thumb with infinite limitsI The product of a finite limit and an infinite limit is infinite if
the finite limit is not 0.
a · ∞ =
{∞ if a > 0
−∞ if a < 0.
a · (−∞) =
{−∞ if a > 0
∞ if a < 0.
I The product of two infinite limits is infinite.
∞ ·∞ =∞∞ · (−∞) = −∞
(−∞) · (−∞) =∞
I The quotient of a finite limit by an infinite limit is zero:
a
∞= 0.
Indeterminate Limits
I Limits of the form 0 · ∞ and ∞−∞ are indeterminate. Thereis no rule for evaluating such a form; the limit must beexamined more closely.
I Limits of the form1
0are also indeterminate.
Indeterminate Limits
I Limits of the form 0 · ∞ and ∞−∞ are indeterminate. Thereis no rule for evaluating such a form; the limit must beexamined more closely.
I Limits of the form1
0are also indeterminate.
Outline
Infinite LimitsVertical AsymptotesInfinite Limits we KnowLimit “Laws” with Infinite LimitsIndeterminate Limits
Limits at InfinityAlgebraic rates of growthExponential rates of growthRationalizing to get a limit
Worksheet
DefinitionLet f be a function defined on some interval (a,∞). Then
limx→∞
f (x) = L
means that the values of f (x) can be made as close to L as welike, by taking x sufficiently large.
DefinitionThe line y = L is a called a horizontal asymptote of the curvey = f (x) if either
limx→∞
f (x) = L or limx→−∞
f (x) = L.
y = L is a horizontal line!
DefinitionLet f be a function defined on some interval (a,∞). Then
limx→∞
f (x) = L
means that the values of f (x) can be made as close to L as welike, by taking x sufficiently large.
DefinitionThe line y = L is a called a horizontal asymptote of the curvey = f (x) if either
limx→∞
f (x) = L or limx→−∞
f (x) = L.
y = L is a horizontal line!
DefinitionLet f be a function defined on some interval (a,∞). Then
limx→∞
f (x) = L
means that the values of f (x) can be made as close to L as welike, by taking x sufficiently large.
DefinitionThe line y = L is a called a horizontal asymptote of the curvey = f (x) if either
limx→∞
f (x) = L or limx→−∞
f (x) = L.
y = L is a horizontal line!
TheoremLet n be a positive integer. Then
I limx→∞
1
xn= 0
I limx→−∞
1
xn= 0
Using the limit laws to compute limits at ∞
Example
Find
limx→∞
2x3 + 3x + 1
4x3 + 5x2 + 7
if it exists.
A does not exist
B 1/2
C 0
D ∞
Using the limit laws to compute limits at ∞
Example
Find
limx→∞
2x3 + 3x + 1
4x3 + 5x2 + 7
if it exists.
A does not exist
B 1/2
C 0
D ∞
SolutionFactor out the largest power of x from the numerator anddenominator. We have
2x3 + 3x + 1
4x3 + 5x2 + 7=
x3(2 + 3/x2 + 1/x3)
x3(4 + 5/x + 7/x3)
limx→∞
2x3 + 3x + 1
4x3 + 5x2 + 7= lim
x→∞
2 + 3/x2 + 1/x3
4 + 5/x + 7/x3
=2 + 0 + 0
4 + 0 + 0=
1
2
Upshot
When finding limits of algebraic expressions at infinitely, look atthe highest degree terms.
SolutionFactor out the largest power of x from the numerator anddenominator. We have
2x3 + 3x + 1
4x3 + 5x2 + 7=
x3(2 + 3/x2 + 1/x3)
x3(4 + 5/x + 7/x3)
limx→∞
2x3 + 3x + 1
4x3 + 5x2 + 7= lim
x→∞
2 + 3/x2 + 1/x3
4 + 5/x + 7/x3
=2 + 0 + 0
4 + 0 + 0=
1
2
Upshot
When finding limits of algebraic expressions at infinitely, look atthe highest degree terms.
Another Example
Example
Find
limx→∞
√3x4 + 7
x2 + 3
SolutionThe limit is
√3.
Another Example
Example
Find
limx→∞
√3x4 + 7
x2 + 3
SolutionThe limit is
√3.
Example
Make a conjecture about limx→∞
x2
2x.
SolutionThe limit is zero. Exponential growth is infinitely faster thangeometric growth
Example
Make a conjecture about limx→∞
x2
2x.
SolutionThe limit is zero. Exponential growth is infinitely faster thangeometric growth
Rationalizing to get a limit
Example
Compute limx→∞
(√4x2 + 17− 2x
).
SolutionThis limit is of the form ∞−∞, which we cannot use. So werationalize the numerator (the denominator is 1) to get anexpression that we can use the limit laws on.
Rationalizing to get a limit
Example
Compute limx→∞
(√4x2 + 17− 2x
).
SolutionThis limit is of the form ∞−∞, which we cannot use. So werationalize the numerator (the denominator is 1) to get anexpression that we can use the limit laws on.
Outline
Infinite LimitsVertical AsymptotesInfinite Limits we KnowLimit “Laws” with Infinite LimitsIndeterminate Limits
Limits at InfinityAlgebraic rates of growthExponential rates of growthRationalizing to get a limit
Worksheet
Worksheet
