MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives
Problem
An unknown function f is defined for all x near x=3, except possibly at x=3. Table 2 gives selected values of f(x).
Table 2: Values of f(x) near x=3
x 2.80 2.90 2.95 3.05 3.10 3.20
f(x) 5.32 5.16 5.08 4.92 4.84 4.68
Assume f is continuous on each side of x=3.
(a) Using only the three data points with x<3, calculate an estimate of x→3−limf(x) and state your reasoning.
(b) Using only the three data points with x>3, calculate an estimate of x→3+limf(x) and state your reasoning.
(c) Combine your answers from parts (a) and (b) to obtain a single numerical estimate for L=x→3limf(x). Justify the method you use to combine the one-sided estimates.
(d) Using only the information provided, give an interval of length 0.2 that you are certain contains L, and explain why your interval is valid.