Consider the function defined for

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

Problem

Consider the function

g(x)=\dfrac{\sin x}{x}

defined for

x>0

. In this problem you will analyze the long–run behavior of

g

. (a) State the largest and smallest possible values of

g(x)

for any real number

x>0

. Justify your answer using properties of the sine function. (b) Explain, without using differentiation or integration, why

\displaystyle \lim_{x\to\infty} g(x)=0

. Your explanation must cite the bounds you found in part (a) and include a clear logical argument leading to the conclusion.