A function is defined for all real numbers by .

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

Problem

A function is defined for all real numbers by

h(x)=x\,e^{-x}\sin\!(x^{2})

. (1) Determine

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

. Provide a rigorous justification that includes an explicit inequality bounding

|h(x)|

. (2) Find

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

. Show all steps in your reasoning. (3) Explain why the two limits in parts (a) and (b) differ, even though both involve factors that approach

0

. Your explanation should refer to the relative rates at which the factors approach their respective limiting values.