A function is defined for all real numbers except by for all x 2

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

Problem

A function

f

is defined for all real numbers except

x=2

\text{for all } x\ne 2,\quad f(x)=\frac{x^2-4}{x-2}.

The value

f(2)

has not yet been specified. Answer the following questions. (a) Identify the type of discontinuity that

f

exhibits at

x=2

and describe—in complete sentences—the difference between this kind of discontinuity and one that is not removable. (b) Determine

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

and state an appropriate value for

f(2)

so that the extended function is continuous at

x=2

. (c) Explain, using the formal definition of continuity in terms of limits, why the value you chose in part (b) eliminates the discontinuity.