1. Section 2.5
Limits at Infinity
Math 1a
October 10, 2007
Announcements
Midterm I is coming: October 24, 7:00-9:00 in Halls A and C

3. Definition
Let f be a function defined on some interval (a, ∞). Then
lim f (x) = L
x→∞
means that the values of f (x) can be made as close to L as we
like, by taking x sufficiently large.

4. Definition
Let f be a function defined on some interval (a, ∞). Then
lim f (x) = L
x→∞
means that the values of f (x) can be made as close to L as we
like, by taking x sufficiently large.
Definition
The line y = L is a called a horizontal asymptote of the curve
y = f (x) if either
lim f (x) = L or lim f (x) = L.
x→∞ x→−∞

5. Definition
Let f be a function defined on some interval (a, ∞). Then
lim f (x) = L
x→∞
means that the values of f (x) can be made as close to L as we
like, by taking x sufficiently large.
Definition
The line y = L is a called a horizontal asymptote of the curve
y = f (x) if either
lim f (x) = L or lim f (x) = L.
x→∞ x→−∞
y = L is a horizontal line!

6. Theorem
Let n be a positive integer. Then
1
lim n = 0
x→∞ x
1
lim =0
x→−∞ x n

8. Using the limit laws to compute limits at ∞
Example
Find
2x 3 + 3x + 1
lim
x→∞ 4x 3 + 5x 2 + 7
if it exists.
A does not exist
B 1/2
C0
D∞

9. Using the limit laws to compute limits at ∞
Example
Find
2x 3 + 3x + 1
lim
x→∞ 4x 3 + 5x 2 + 7
if it exists.
A does not exist
B 1/2
C0
D∞

10. Solution
Factor out the largest power of x from the numerator and
denominator. We have
2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 )
=3
4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 )
2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3
lim = lim
x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3
2+0+0 1
= =
4+0+0 2

11. Solution
Factor out the largest power of x from the numerator and
denominator. We have
2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 )
=3
4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 )
2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3
lim = lim
x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3
2+0+0 1
= =
4+0+0 2
Upshot
When finding limits of algebraic expressions at infinitely, look at
the highest degree terms.

12. Another Example
Example
Find √
3x 4 + 7
lim
x2 + 3
x→∞

14. Another Example
Example
Find √
3x 4 + 7
lim
x2 + 3
x→∞
Solution √
The limit is 3.

15. Example
x2
Make a conjecture about lim .
x→∞ 2x

17. Example
x2
Make a conjecture about lim .
x→∞ 2x
Solution
The limit is zero. Exponential growth is infinitely faster than
geometric growth

18. Rationalizing to get a limit
Example
4x 2 + 17 − 2x .
Compute lim
x→∞

20. Rationalizing to get a limit
Example
4x 2 + 17 − 2x .
Compute lim
x→∞
Solution
This limit is of the form ∞ − ∞, which we cannot use. So we
rationalize the numerator (the denominator is 1) to get an
expression that we can use the limit laws on.

21. Worksheet