Mathematics

Problem

Consider the function

f(x)=x^{2}\cos\bigl(\tfrac{5}{x}\bigr)

defined for

x\neq 0

and extended by

f(0)=0

. a. Establish an inequality that relates

|f(x)|

to a simpler algebraic expression for all

x\neq 0

. State the trigonometric fact you rely on. b. Using the inequality from part (a), determine

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

. Present a complete justification for your conclusion. c. Suppose instead that

f_k(x)=x^{2}\cos\bigl(\tfrac{k}{x}\bigr)

, where

k

is a non-zero constant. Determine

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

in terms of

k

and justify your result.