Notes: 2.5 Infinite Limits
4.5 Limits at Infinity
AP Calculus AB
Limits Involving Infinity
Graphs with vertical asymptotes:
Vertical Asymptote -
Example 1: Find each limit.
a.
b.
c.
The function f(x) will have a vertical asymptote at x = a if we have
any of the following limits at x = a:
It is only required that you have one of the above limits for a
function to have a vertical asymptote at x = a.
Limits Approaching Infinity
and
Graphs with horizontal asymptotes:
Example 2: Find the limit.
Verification of Horizontal Asymptotes
The function f(x) will have a horizontal asymptote at y = L if either of the
following are true:
OR
Example 3
For each function:
a. Find the vertical and horizontal asymptotes.
b. Find the limit as x approaches the vertical asymptote(s) from the left
and from the right.
c. Show that the horizontal asymptote is correct by finding the limits as
x approaches negative and positive infinity.
1)
For each function:
a. Find the vertical and horizontal asymptotes.
b. Find the limit as x approaches the vertical asymptote(s) from the left
and from the right.
c. Show that the horizontal asymptote is correct by finding the limits as
x approaches negative and positive infinity.
2)
Example 4: Find each limit.
a.
b.
c.
d.
End Behavior
For some functions, especially numerically large values of x, the end
behavior of a complicated function can be modeled by a simpler
function that acts virtually in the same way.