Define the piecewise function f(x)= cases 1- x x^ 2 , & x0 4, & x=0 cases Determine…

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

Problem

Define the piecewise function

f(x)=\begin{cases}\dfrac{1-\cos x}{x^{2}}, & x<0\\ \dfrac{\ln(1+x)}{x}, & x>0\\ 4, & x=0 \end{cases}

Determine the type of discontinuity of

f

x=0

. (a) Removable discontinuity, because the two-sided limit exists but differs from

f(0)

. (b) Infinite discontinuity, because

f(x)

grows without bound as

x\to0

. (c) Jump discontinuity, because the one-sided limits are finite but unequal. (d) No discontinuity; the function is continuous at

x=0