1. 2.2 Limits Involving Infinity
Created by Greg Kelly, Hanford High School, Richland, Washington
Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

2. Limits outward “toward infinity”
As the denominator gets larger (as x →∞),
the overall fraction value gets smaller.
There is a horizontal asymptote if either: or

3. Adding 1 becomes insignificant as .
Example 1:
Adding 1 becomes insignificant as
There is a horizontal asymptote at a
height limit of 1.

4. When we graph this function, the limit appears to be zero.
Find:
Example 2:
When we graph this function, the limit appears to be zero.
so for :
by the Sandwich Theorem:

5. Example 3:
Find:

6. Limits that are infinite in height
As the denominator approaches zero,
the magnitude of the fraction gets
very large.
vertical asymptote at x=0.
If the denominator is positive, then the fraction’s overall sign is positive.
If the denominator is negative, then the fraction’s overall sign is negative.
The overall limit as x approaches zero does not exist,
since the left-hand and right-hand limits don’t match.

7. The overall limit as x approaches zero is
Example 4:
The denominator is positive after squaring x, so the limit has the same sign from left and right.
The overall limit as x approaches zero is
“positive without bound,” since the left-hand and right-hand
limits both trend toward positive infinity.

8. End Behavior Models
An end behavior model describes the height behavior of a function as x approaches positive infinity or
as x approaches negative infinity.
A function g(x) is:
a right-end-behavior model for f if and only if
a left-end-behavior model for f if and only if

9. Find right- and left-end models for
Example 7:
Find right- and left-end models for
As , approaches zero...
whereas x is further
from the x-axis, so x dominates on the right.
becomes a right-end behavior model.
Test of
model
As , is much further away from the x-axis,
therefore is dominant on the left.
becomes a left-end behavior model.
Test of
model
Our model
is correct!

10. becomes a right-end behavior model.
Example 7:
becomes a right-end behavior model.
becomes a left-end behavior model.
On your calculator, compare which pairs of graphs
match the best on the right and on the left of the y-axis:
Window:

11. Example:
For rational functions, the end behavior model comes from the highest power terms
in the numerator and denominator:
f(x) ≈
when |x| large
dominant terms in numerator and denominator

12. You can adjust the direction of limits:
“outbound” limit for an input
equals
“inbound” limit (toward x = 0) for the input’s reciprocal p