Consider the function for .

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

Problem

Consider the function

h(x)=\dfrac{\sin(1/x)}{x}

for

x\neq 0

. (a) Describe qualitatively how

h(x)

behaves as

x

approaches

0

. Focus on the oscillatory nature and the magnitude of the function. (b) Provide a rigorous argument demonstrating that

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

does not exist. Your explanation should exhibit at least two sequences

\{x_n\}

and

\{y_n\}

, each tending to

0

, such that

h(x_n)

and

h(y_n)

approach different limits.