Limits at Infinity

Limits at infinity are used to describe the behavior of functions as the independent variable increases or decreases without bound. If a function approaches a numerical value L in either of these situations, write

and f( x) is said to have a horizontal asymptote at y = L. A function may have different horizontal asymptotes in each direction, have a horizontal asymptote in one direction only, or have no horizontal asymptotes.
Evaluate 1: Evaluate
Factor the largest power of x in the numerator from each term and the largest power of x in the denominator from each term.
You find that


The function has a horizontal asymptote at y = 2.
Example 2: Evaluate
Factor x³ from each term in the numerator and x⁴ from each term in the denominator, which yields


The function has a horizontal asymptote at y = 0.
Example 3: Evaluate .
Factor x² from each term in the numerator and x from each term in the denominator, which yields


Because this limit does not approach a real number value, the function has no horizontal asymptote as x increases without bound.
Example 4: Evaluate .
Factor x³ from each term of the expression, which yields
As in the previous example, this function has no horizontal asymptote as x decreases without bound.

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By Muhammad Atif Saeed on 01:55. Filed under Calculus , feature , Mathandstat . Follow any responses to the RSS 2.0. Leave a response