Let S( )= 1+ , -1 ^ 2 for and defined so that is continuous at the origin.

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

Problem

Let

S(\theta)=\frac{\sqrt{1+\theta\,\sin\theta}-1}{\theta^{2}}

for

\theta\neq 0

and

S(0)

defined so that

S

is continuous at the origin. (a) Determine

\displaystyle\lim_{\theta\to 0}S(\theta)

and explain each algebraic or trigonometric simplification used. (b) Using your result from part (a), assign a value to

S(0)

so that

S

becomes continuous at

\theta=0

. (c) Prove that

S

is differentiable at

\theta=0

and find

S'(0)