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MATH 1830
Section 2.2
Limits and Asymptotes
*Infinite Limits and Vertical Asymptotes
Page 124
Example 1
Analyzing a Limit
Page 125
Example 2
Analyzing an Infinite Limit
Infinite Limits and Vertical Asymptotes
The line x = a is a vertical asymptote of the graph of f if any of the following are true..
 If as xa the function values f(x) increase or decrease without bound,
that is, lim f ( x)   or lim f ( x)   , then x = a is a vertical asymptote.
xa
x a
 If as xa the function values f(x) increase or decrease without bound,
that is, lim f ( x)   or lim f ( x)   , then x = a is a vertical asymptote.
x a
xa
NOTE: If both the left- and the right-hand limits exhibit the same behavior, we say that
lim f ( x)   (or ).
xa
NOTE: These limits do not technically exist, since   are not real numbers, they are
merely concepts. These symbols are being used to describe the behavior of the
function near the vertical asymptote.
*Limits at Infinity and Horizontal Asymptotes
Page 128
Example 5
Analyzing Limits at Infinity (exponential growth)
Page 129
Example 6
Applying Limits at Infinity (exponential decay)
Page 131
Example 7
Determining Horizontal Asymptotes
Limits at Infinity and Horizontal Asymptotes – For any function f, if lim f ( x)  L ,
x 
then the line y = L is a horizontal asymptote for the graph of f.
Special Limits at Infinity
1
0
x  x n
1
2. For n, a positive real number, lim n  0 , provided that x n is a real number
x  x
for negative values of x.
1. For n, a positive real number, lim