MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives
Problem
A function f is displayed on a graphing calculator. The default viewing window is −10≤x≤10 and −10≤y≤10. In that window the trace cursor suggests that the graph of f is nearly a straight line passing through the point (3,5), and the plotted points on either side of x=3 appear to lie on that same line.
To investigate further, a student collects numerical output from the calculator using the TABLE feature. The values of f(x) for inputs close to x=3 are shown below.
x
f(x)
2.900
4.997
2.990
4.999
2.999
4.996
3.001
5.004
3.010
5.001
3.100
5.003
However, when the student changes the window to 2.99≤x≤3.01 and 4.9≤y≤5.1, the graph no longer looks linear. Instead, it oscillates rapidly above and below y=5 many times between x=2.99 and x=3.01.
Use the information above to analyze x→3limf(x).
(a) Based only on the appearance of the graph in the default window and the original TRACE reading at (3,5), state the limit value one might initially report for x→3limf(x) and briefly explain that choice.
(b) Using the numerical data in the table and the observations from the zoomed-in window, determine whether x→3limf(x) actually exists. If it exists, give its value; if it does not exist, provide a convincing argument based on the evidence.
(c) Describe in detail how the scale of the original window masked the true behavior of f near x=3. In your explanation, reference both the graphical and numerical information provided.