Define the function g(x)=x^ 3 ! ( 1x )+x^ 2 ! ( 1 x^ 2 )

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

Problem

Define the function

g(x)=x^{3}\sin\!\Bigl(\tfrac1x\Bigr)+x^{2}\cos\!\Bigl(\tfrac1{x^{2}}\Bigr),\qquad x\ne 0,

and set

g(0)=0

. (1) For

x\ne 0

, establish an explicit numerical bound of the form

|g(x)|\le k\,|x|^{2}

, where

k

is a constant you must determine. Show each step leading to the inequality. (2) Using the bound from part (a), evaluate

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

. Provide full justification that the limit you obtain is correct.