For , define…

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

Problem

For x≠0x \neq 0, define hh by   h(x)=xsin⁡ ⁣(1x)+e−xcos⁡x.h(x) = x\sin\!\left(\frac{1}{x}\right) + e^{-x}\cos x. (a) Show that lim⁡x→∞xsin⁡ ⁣(1x)=1\displaystyle \lim_{x\to\infty} x\sin\!\left(\tfrac{1}{x}\right) = 1. Include a sound justification. (b) Determine lim⁡x→∞e−xcos⁡x\displaystyle \lim_{x\to\infty} e^{-x}\cos x. (c) Using the results from parts (a) and (b), evaluate lim⁡x→∞h(x)\displaystyle \lim_{x\to\infty} h(x). (d) Evaluate lim⁡x→−∞h(x)\displaystyle \lim_{x\to -\infty} h(x). Discuss any oscillatory complications and how they influence the limit. (e) Summarize all horizontal asymptote(s) of y=h(x)y = h(x) and provide a rigorous justification for each. Clearly state whether the function approaches the same or different values from the left and right extremes of the xx-axis.

Related theory

Browse the problem library