MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives
Problem
For a certain function g, two separate screenshots from a graphing utility are described.
Screen A: The viewing window is −2π≤x≤2π and −4≤y≤4. Near x=0 the curve appears smooth and seems to cross the y–axis at approximately y≈0.7.
Screen B: The viewing window is −0.5≤x≤0.5 and −1≤y≤1. In this window, the portion of the graph between x=−0.1 and x=0.1 shows a dense collection of tiny oscillations whose amplitude remains below ∣y∣=0.8, but the oscillations become more frequent as x approaches 0.
A small numerical sample generated for g is given below.
x
g(x)
-0.10
0.738
-0.05
0.714
-0.02
0.701
-0.01
0.698
0.01
0.702
0.02
0.699
0.05
0.717
0.10
0.740
Consider the limit x→0limg(x).
(a) Indicate which screenshot (A or B) gives stronger evidence about the existence of the limit of g as x approaches 0. Provide a brief justification based on specific graphical features.
(b) Using the more informative screenshot together with the numerical data, determine whether x→0limg(x) exists. If it exists, estimate its value to three decimal places and justify your reasoning.
(c) Explain why the conclusion in part (b) could not have been made confidently from Screen A alone.