A function is defined by p(x)= cases x, & x2

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

A function

p

is defined by

p(x)=\begin{cases} \sin x, & x<2,\\ 3, & x=2,\\ \cos x, & x>2. \end{cases}

Address the following regarding the point

x=2

. (a) Determine

\displaystyle\lim_{x\to2}p(x)

and compare it to

p(2)

. Explain fully whether

p

is continuous at

x=2

. (b) Without performing detailed derivative calculations, give a reasoned argument as to whether an instantaneous rate of change can exist for

p

x=2

. Refer explicitly to appropriate limit or continuity considerations in your explanation.