A function is defined for all in a small neighborhood of
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A function p is defined for all x=0 in a small neighborhood of x=0. Its graph near the origin is described by the following observations taken from a grid whose horizontal axis is scaled normally (each grid square represents one unit) while the vertical axis is magnified so that one vertical grid square represents 0.1 unit on the y-axis.
• As x approaches 0 from either side, the plotted points lie between y=−0.06 and y=−0.04.
• The point (0,−0.05) is shown as an open circle.
• For x<−0.1 and for x>0.1, the graph gradually moves away from the x-axis, but it never crosses it.
Answer the following questions about p.
(a) Estimate x→0−limp(x) using the information given. Explain how the vertical scaling affects your estimate.
(b) Estimate x→0+limp(x) using the information given. Explain how the vertical scaling affects your estimate.
(c) Based on parts (a) and (b), determine x→0limp(x), or state that the limit does not exist. Provide a clear justification that references both sides of x=0 and the scaling described.
(d) Suppose further that the exact functional rule for p is unknown but smooth everywhere except possibly at x=0. Describe at least one additional piece of information that would definitively confirm the limit you stated in part (c).