A twice–differentiable function is described graphically as follows
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A twice–differentiable function g is described graphically as follows.
• As x→∞, the graph approaches the horizontal line y=3 from below, with the distance between the curve and the line decreasing monotonically.
• As x→−∞, the graph bends downward, flattening out and appearing to level off near the line y=−2 but never quite reaching it.
• No other information about g is given.
Use the description to analyze the end behavior of g.
(a) Determine x→∞limg(x) and explain, in words, how the stated graphical behavior supports your conclusion.
(b) Determine x→−∞limg(x) and provide reasoning that connects the verbal description of the left-hand tail to your limit statement.
(c) Describe how the limits found in parts (a) and (b) dictate the presence or absence of horizontal asymptotes for the graph of g. Justify your answer without performing any algebraic computation.