Let . Answer the following questions concerning the behavior of near…

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

Problem

Let

f(x)=\dfrac{\sin(3x)}{x}

. Answer the following questions concerning the behavior of

f

near

x=0

. (a) Rewrite

f(x)

so that any constant multiple of

x

that appears inside a trigonometric function is separated from

x

itself. Express the result as a product of two factors, one of which approaches

1

x\to 0

. Do not evaluate any limits in this step. (b) Using your expression from part (a), demonstrate that

\displaystyle \lim_{x\to 0} f(x)

exists and determine its exact value. (c) Describe in words what the value you found in part (b) tells you about the local slope of

\sin(3x)

x=0

. Justify your description briefly.