Mathematics

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