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AP Calculus AB – Infinite Limits and Finding Asymptotes
What is an Infinite
Limit?
When a function, f, INCREASES without bound or DECREASES without
bound as x approaches a ‘target’ value, c, then the limit is called an
INFINITE LIMIT. The notation is:
OR
This does NOT mean that the limit exists and is equal to . Instead it is
expressing that the limit FAILS to exist at x -> c because of its
unbounded behavior.
Examples:
Definition of a
Vertical Asymptote
If f(x) approaches infinity (or negative infinity) as x approaches c from
right or left, then the line x = c is a vertical asymptote of the graph f.
A vertical asymptote occurs when the denominator of a rational function
equals 0 at a particular value of x.
h(x) = f(x)/g(x), and f and g are both continuous on an open interval that
contains c. If f(c) 0 (numerator), and g(c) = 0 (denominator), then h(x) a
has a vertical asymptote at x = c.
Examples
NOTE: The numerator MUST be NON-Zero at c.
If BOTH are = 0, then it is INDETERMINATE FORM (0/0)
Example:
Both 2 and -2 will make the denominator zero, but 2 also makes the
numerator zero, so x=2 is NOT and asymptote (see the graph to confirm).
Using the ‘divide out’ method, you can factor the numerator and
denominator to get f(x) = (x+4)/(x+2). Then you can see that:
and
Properties of
Infinite Limits
If
and
1. Limit of sum or difference =
2. Limit of product =
if L > 0 or -
if L < 0.
3. Limit of g(x)/f(x) = 0 (since a fraction with a constant in the
numerator and an infinitely large number in the denominator
approaches 0.