$axy,|sinx-siny||x-y|.$

$bx,yxy|sinx-siny|>k|x-y|f(0,)\underset{x}{lim}f(x)=1\underset{x}{lim}{\displaystyle \underset{x}{\overset{x+2}{}}}tsin(\frac{3}{t})f(t)dt0,$ then there exist $x,y$ with $xy$ such that $|sinx-siny|>k|x-y|.$

Let $fbe$ differentiable $on(0,).$ Assume that $\underset{x}{lim}f(x)=1.$ Prove that the following $limit$ exists, and find

its value:

$\underset{x}{lim}{\displaystyle \underset{x}{\overset{x+2}{}}}tsin(\frac{3}{t})f(t)dt$