Limits at Infinity
Definition of a limit as x → ±∞
Let f be a function defined on some interval (a, ∞) ((−∞, a), resp.).
Then
lim f (x) = L
lim = L, resp.
x→∞
x→−∞
means that the values of f (x) can be made arbitrarily close to L by
taking x sufficiently large (large negative, resp.).
Example: f (x) = 1/x
By taking x arbitrarily large and positive, f (x) becomes arbitrarily close
to 0. Hence limx→∞ (1/x) = 0.
By taking x arbitrarily large and negative, f (x) becomes arbitrarily close
to 0. Hence limx→−∞ (1/x) = 0.
Limits at infinity from graphs
Below is the graph of a function f :
limx→∞ f (x) = 2
limx→−∞ f (x) = − 1
limx→0 f (x) = − ∞
limx→2− f (x) = − ∞
limx→2+ f (x) = ∞
limx→2 f (x) = DNE
How to calculate limits at infinity?
Recall that we used knowledge of simple limits such as limx→a c = c and
limx→a x = a, together with the limit laws and algebra to find
complicated limits. The same is true now.
Use limit laws (almost all are still valid in this case), some algebra, and
the following theorem
Theorem
If r > 0 is a rational number, then
lim
x→∞
1
=0
xr
If r > 0 is a ration number such that x r is defined for all x (e.g.
r = 1/3), then we also have
lim
x→−∞
1
=0
xr
Example
A tank contains 5000 L of pure water. Brine that contains 30 g of salt
per liter of water is pumped in the tank at a rate of 25 L / min. The
concentration of salt after t minutes (in grams per liter) is given by:
C (t) =
30t
200 + t
What happens to the concentration of salt as time goes on?
To find the answer, we take the limit of the concentration function as
t → ∞. First using algebra,
lim C (t) = lim
t→∞
t→∞
30t
30t
= lim
= lim
t→∞
t→∞ t( 200 + 1)
200 + t
t
30
+1
200
t
Recall that one of our limit laws states that if lim g (x) and lim f (x) exist
AND lim g (x) 6= 0, then lim[f (x)/g (x)] = lim f (x)/ lim g (x).
Example (cont.)
Let’s check that we can use it, using other limit laws:
lim 30 = 30
x→∞
lim (
x→∞
200
1
200
+ 1) = lim
+ lim 1 = 200 lim + 1 = 0 + 1 = 1
x→∞
x→∞
x→∞
t
t
t
So both limits exist and the limit of the denominator is not zero and
lim C (t) = lim
t→∞
30
limt→∞ 30
30
=
=
= 30
1
+1
limt→∞ ( 200
+
1)
t
t→∞ 200
t
Let’s also check that this agrees with our intuition. As time goes on, the
proportion of the total volume of water that is brine will increase. The
original amount of pure water will become small when compared with the
amount of brine. Thus the concentration of the whole mixture will
approach that of the brine, i.e. 30 grams per liter.
A limit at infinity that does not exist
Example: f (x) = sin(x)
There no single (unique) number L such that f can be made arbitrarily
close to L as x gets arbitrarily large. Hence limx→∞ f (x) does not exist.
Compare this to the limit limx→0 sin(1/x)
Precise definition (the case of x → ∞)
Precise definition (the case of x → ∞)
Let f be a function defined on (a, ∞). Then
lim f (x) = L
x→∞
means that for every > 0, there is a corresponding number N such that
if x > N then |f (x) − L| < Infinite Limits
Infinite limits
Let f be a function defined on both sides of a, except possibly at a itself.
Then
lim f (x) = ∞
lim f (x) = −∞
x→a
x→a
means that the values of f (x) can be made arbitrarily large (resp. large
negative) by taking x sufficiently close to a, but not equal to a. [Leftand right-hand infinite limits are defined analogously.]
Example: f (x) = 1/x. Then limx→0+ (1/x) = ∞ and
limx→0− (1/x) = −∞
Infinite Limits
Infinite limits at infinity
The notation limx→∞ f (x) = ∞ (resp. −∞) means that the values of
f (x) become large (resp. large negative) as x becomes large.
Infinite limits at negative infinity
The notation limx→−∞ f (x) = ∞ (resp. −∞) means that the values of
f (x) become large (resp. large negative) as x becomes large negative.
Examples:
lim x = ∞
x→∞
lim x = −∞
x→−∞
lim (−x 3 ) = −∞
x→∞
lim (−x 3 ) = ∞
x→−∞
Example
Evaluate limx→∞ (x −
√
x).
You might want to try
√
√
lim (x − x) = lim (x) − lim ( x) = ∞ − ∞
x→∞
x→∞
x→∞
√
but this is NOT valid, because limx→∞ (x) = limx→∞ ( x) = ∞ which
are not number, and hence the limit laws don’t apply. Instead, try
factoring out the term with the largest exponent:
lim (x −
x→∞
√
1
x) = lim x(1 − √ )
x→∞
x
Now the first factor x becomes arbitrarily large as x → ∞, while the
second factor (1 − √1x ) approaches 1 as x → ∞. Thus the product must
become arbitrarily
large as x → ∞, and we can write
√
limx→∞ (x − x) = ∞
Asymptotes
Definition of Vertical Asymptote
The line x = a is called a vertical asymptote of the curve y = f (x) if at
least one of the following statements is true:
limx→a f (x) = ∞
limx→a− f (x) = ∞
limx→a+ f (x) = ∞
limx→a f (x) = −∞ limx→a− f (x) = −∞ limx→a+ f (x) = −∞
Definition of Horizontal Asymptote
The line y = L is called a horizontal asymptote of the curve y = f (x) if
either
lim f (x) = L
x→∞
or
lim f (x) = L
x→−∞
Example: The function f (x) = 1/x has a vertical asymptote x = 0 and a
horizontal asymptote y = 0.
Physical Example of Vertical Asymptote
In the theory of relativity, the mass of a particle is
m= p
m0
1 − v 2 /c 2
where m0 is the rest mass of the particle, m is the mass when the particle
moves with speed v relative to the observer, and c is the speed of light.
What happens to the mass as the particle velocity approaches the speed
of light?
p
Mathematically speaking, the question is: find limv →c − m0 / 1 − v 2 /c 2 .
The numerator remains some fixed number m0 > 0, while v 2 /c 2 < 1, so
the denominator
p approaches 0. Hence v = c is a vertical asymptote and
limv →c − m0 / 1 − v 2 /c 2 = ∞. Physically speaking, your mass becomes
infinite as your velocity approaches the speed of light. So you can never
go faster than the speed of light!