A student is reviewing the long–term behavior of continuous functions
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A student is reviewing the long–term behavior of continuous functions.
The function f is defined for all real numbers and it is known that
x→∞limf(x)=3andx→−∞limf(x)=−2.
Answer the following questions about the relationship between limits at infinity and horizontal asymptotes.
(a) State a precise mathematical definition of the statement “the line y=L is a horizontal asymptote of the graph of f as x→∞."
(b) Using your definition from part (a), explain why the given limit information implies that the graph of f has two distinct horizontal asymptotes.
(c) Write the equations of all horizontal asymptotes of f and describe, in words, how the graph of f behaves as x becomes very large in the positive and negative directions.