A technology‐based exploration is carried out on a function that is known to be…
MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions
Problem
A technology‐based exploration is carried out on a function q that is known to be continuous everywhere except possibly at x=5. The calculator’s default plot shows an apparently flat segment near x=5 at height y=2.1. Concerned about hidden behavior, a student records the following values of q.
x
q(x)
4.90
2.109
4.95
2.105
4.99
2.154
5.01
2.047
5.05
2.095
5.10
2.102
The student then graphs q on a dramatically enlarged window: 4.99≤x≤5.01 and 2.0≤y≤2.3. In that view the graph wiggles rapidly, crossing the horizontal line y=2.1 many times.
Answer the questions below about x→5limq(x).
(a) Using the numerical data, estimate the left‐hand limit and the right‐hand limit of q as x approaches 5. Clearly state each estimate.
(b) Interpret the behavior seen in the magnified window to refine or revise the estimates from part (a).
(c) Determine, with justification, whether x→5limq(x) exists. Base your argument on both numerical and graphical evidence.
(d) Design a systematic investigative strategy (including specific window settings or step sizes) that another student could follow to verify your conclusion independently. Explain the rationale for each element of your strategy.