As for the typesetting, we use LaTeX. There is a LaTeX subforum here were you can get some assistance.

As for the question, the definition of a multivariable limit is that for all $\displaystyle \begin{align*} \epsilon > 0 \end{align*}$ there exists a $\displaystyle \begin{align*} \delta > 0 \end{align*}$ such that $\displaystyle \begin{align*} \sqrt{ \left( x - a \right) ^2 + \left( y - b \right) ^2 } < \delta \implies \left| f(x,y) - L \right| < \epsilon \end{align*}$. Then $\displaystyle \begin{align*} \lim_{(x,y) \to (a,b)} f(x,y) = L \end{align*}$.

So in this case $\displaystyle \begin{align*} f(x,y) = \frac{x^2\,y\,\left( y - 1 \right) ^2 }{x^2 + \left( y-1 \right) ^2 }, a = 0, b = 1, L = 0 \end{align*}$. Go from here.