A function is defined for all by p(x)= ( 1 x-2 ) x-2 +4

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

A function

p

is defined for all

x\ne 2

p(x)=\frac{\sin\left(\tfrac{1}{x-2}\right)}{x-2}+4.

When graphed on a calculator in the window

-5 \le x \le 5

and

0 \le y \le 8

p

appears to approach the horizontal line

y=4

x

approaches

2

. Analyze the behavior of

p

near

x=2

and decide whether

\displaystyle \lim_{x\to 2} p(x)

exists. (a) Using the definition of

p

, rewrite

p(x)-4

as a product of two factors. State the two factors explicitly. (b) Explain why one of the factors identified in part (a) becomes unbounded as

x\to 2

while the other remains bounded. Support your explanation with appropriate limit properties or theorems. (c) Prove that

\displaystyle \lim_{x\to 2} p(x)

does not exist, even though the calculator graph in the given window suggests convergence to

4

. (d) Discuss how successively smaller

x

–intervals centered at

x=2

would change the visual appearance of the graph and what a student should look for to avoid an incorrect conclusion about the limit.