A function satisfies the following table of values near

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

Problem

A function

f

satisfies the following table of values near

x=1

$x$	0.8	0.9	0.99	1.01	1.1	1.2
$f(x)$	2.2	2.4	2.95	5.05	5.6	5.8

(a) Estimate

\displaystyle \lim_{x\to 1^-} f(x)

and

\displaystyle \lim_{x\to 1^+} f(x)

, supporting each estimate with explicit numerical evidence. (b) Determine whether

\displaystyle \lim_{x\to 1} f(x)

exists. Provide a clear justification grounded in the relationship between one–sided limits and two–sided limits. (c) Suppose

f

is redefined at

x=1

so that

f(1)=4.0

. Classify the resulting discontinuity at

x=1

as removable, jump, or neither, and justify your classification. (d) Discuss whether

f

could possess a derivative at

x=1

under the new definition in part (c). Base your argument on the structure of

f

's limiting behavior.