\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\usepackage{epsfig}
\begin{document}
\parindent=0pt

\begin{center}
\textbf{Partial Differentiation}

\textit{\textbf{Limits}}
\end{center}

\textbf{Question}

Evaluate the given limit. If the limit does not
exist, explain why.

$$\lim_{(x,y)\to(0,0)} \frac{x^2+y^2}{y}$$


\textbf{Answer}

$ \lim_{(x,y)\to(0,0)} \frac{x^2+y^2}{y}$ does not exist.

If $(x,y)\to (0,0)$ along $x=0$ then $\ds \frac{x^2+y^2}{y}=y \to 0$

If $(x,y)\to (0,0)$ along $y=x^2$ then $\ds \frac{x^2+y^2}{y}=1+x^2
\to 1$

\end{document}