A differentiable function is defined for all in an open interval containing
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A differentiable function p is defined for all x in an open interval containing x=1, except possibly at x=1. Table 1 lists several sampled values of p(x).
Table 1: Sampled values of p(x)
x 0.50 0.75 0.90 1.10 1.25 1.50
p(x) 2.436 2.112 2.028 1.972 1.894 1.772
The behavior of p(x) as x approaches 1 is of interest.
(a) Identify and describe the trend you observe in p(x) as x approaches 1 from both the left and the right.
(b) Explain, with reference to the data in Table 1, why using values of x that are closer to 1 is expected to yield a more accurate numerical estimate of x→1limp(x).
(c) Suppose additional measurements taken for x values within 0.02 of 1 are subject to independent measurement noise of magnitude at most 0.01. Discuss qualitatively how this noise could influence the accuracy of a limit estimate based on those measurements, and propose at least one strategy to mitigate its effect.