$($i$)$ formulate a precise statement of what is meant by

$\underset{x}{lim}f(x)=L$

and use it to show that

$\underset{x}{lim}\frac{1}{x}=0.$

$($ii$)$ show that a function defined by

$f(x)=\{\begin{array}{cc}sin(\frac{1}{x^2}),& x0\\ 0,& x=0\end{array}$

is not continuous at $0.$