Mathematics

Problem

Evaluate the limit

\lim_{x\to 0}\bigl(x\sin\tfrac{1}{x}+2x^{3}\cos\tfrac{1}{x^{2}}\bigr).

(a) Establish appropriate inequalities that bound

|x\sin\tfrac{1}{x}|

and

|2x^{3}\cos\tfrac{1}{x^{2}}|

for all

x\neq 0

. (b) Use the inequalities from part (a) to determine the overall limit. Provide a clear, logically ordered argument.