Let be differentiable on an open interval containing with and

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

f

be differentiable on an open interval containing

x=c

with

f(c)=3

and

f'(c)\neq 0

. Consider

h(x)=\dfrac{1}{|f(x)-3|}

. Which statement about

h

x=c

is most accurate? (a)

h

has a removable discontinuity at

x=c

because

\displaystyle\lim_{x\to c}h(x)

exists and is finite. (b)

h

has an infinite (non-removable) discontinuity at

x=c

because

|f(x)-3|\to 0

h(x)\to \infty

from both sides. (c)

h

has a jump discontinuity at

x=c

because the sign of

f(x)-3

changes, so the one-sided limits of

h

are different. (d)

h

has an infinite discontinuity, but the left-hand limit is

-\infty

while the right-hand limit is

+\infty

because

f(x)-3

changes sign even inside the absolute value.