Math 175, Sp16
Notes and Learning Goals
Lesson 3-3
Limits at Infinity
Part I: Vocabulary and Notation
• The notation for limit at infinity is lim f (x).
x→∞
You read this: limit of f (x) as x goes to1 infinity.
• There are three basic types of behavior for a function, f (x), as x approaches infinity.
Know how to recognize each type and know how to write the notation for it.
1. Converges to a Number.
Notation is:
lim f (x) = L
Convergent Examples:
x→∞
where L is a number.
We say f has a horizontal asymptote at L.
Divergent Examples:
2. Diverges to Infinity.
Notation is:
lim f (x) = ∞
x→∞
Or
lim f (x) = −∞
x→∞
3. Everything Else.
We say the limit does not exist.
Notation is
lim f (x) DN E
One possible DNE Example:
x→∞
1
Or “approaches”
1
Part II: Speed to Infinity
• If two functions both diverge to infinity, it is possible that one does so faster than the other.
Know how to tell which is faster, or if they are the same speed. Also know the notation:
– If f is faster than g the notation is f g or g f .
– If they are the same speed the notation is f ∼ g.
• Be able to quickly decide the relative speeds of any of the following simple functions: logarithms,
roots, powers, and exponentials. Here are the essential facts.
– Roots are just fractional powers. Lower powers are slower than higher powers.
x1/2 x3 x5
– Any logarithm is slower than any power, including fractional powers.
√
log10 x x
– Any power is slower than any exponential.
x100 2x
– Lower base exponentials are slower than higher base exponentials.
2x ex 10x
– All logs are the same speed.
log10 x ∼ log2 x
• Be able to rank any collection of simple functions by speed. You should not have to do any algebra
or calculus. Just write down the answer.
√
Example: Rank by speed. x2 , 2x , ln x, x.
√
Solution: ln x x x2 2x
• There is also an algebraic technique. To compare f and g, assuming both diverge to infinity,
compute the limit of f /g.
f
= 0 then f g.
– If lim
x→∞
g
f
– If lim
= ∞ then f g.
x→∞
g
f
– If lim
= any non-zero number, then f ∼ g.
x→∞
g
2
f
• Be able to compute lim
. L’Hôpital’s Rule will work. (The link includes videos.)
x→∞
g
• You should also know how to find the limit using dominant terms of f and g. If f and g have
more than one term you can ignore all but the fastest term of each.
4x2 − 5x + 1
4x2
=
lim
= −4
x→∞ x − x2 + 2
x→∞ −x2
Example: lim
x5 − 5x
−5x
=
lim
=∞
x→∞ 2x − 3x
x→∞ −3x
(Because the x’s cancel.)
(Because 5x is faster than 3x .)
Example: lim
Part III: Tail Thickness
• If f (x) diverges to infinity, then the function
1
will have a horizontal asymptote at zero. It
f (x)
will look like this:
The far right of the graph is called the tail.2
• Be able to tell if two tails are different, and if so, which is thicker. For simple functions this is
just as easy as speed to infinity:
If f is faster than g, then
1
1
has a thinner tail than .
f
g
As with fractions, bigger denominators mean smaller fractions: 4 is bigger than 2, so
• Even the notation is the same. If f g, then the notation for tail thickness is
1
1
f
g
2
Sometimes it is referred to as an asymptotic tail.
3
1
4
< 12 .
• Be able to rank any collection of simple functions by tail thickness. You should not have to do
any algebra or calculus. Just write down the answer, based on the speed of the denominators.
1
1
Example: Which has a thicker tail, x or 3 ?
2
x
1
1
(Because 2x is faster than x3 .)
Solution: x 3
2
x
• Be able to replace a complicated function with a simple function that has the same tail thickness.
You typically use these two facts:3
– Constant multiples do not affect speed to infinity or tail thickness.
– You can rely on dominant terms in a quotient.
Example: Find the simplest function whose tail is the same thickness as the tail of
4x3 + 2x + 1
3x5 + 4x3 + 7x
Solution:
4x3 + 2x + 1
4x3
∼ 5
3x5 + 4x3 + 7x
3x
(locate dominant terms)
∼
x3
x5
(ignore constant multiples)
∼
1
x2
(simplify)
3
The mathematically exact method is limit comparison, as with speed to infinity. These two tricks cover most limit
situations. However, more complicated examples may need L’hôpital’s rule or even more advanced limit techniques.
4