\documentclass{article}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{enumerate}
\usepackage{tikz}
\usepackage{circuitikz}
\usepackage{mathrsfs}

\begin{document}
\flushleft\underline{Homework 6 --- MATH 1586 Spring 2018} \\
Recall that $z=f(x,y)$ describes a surface. The level curves of the surface are the curves we get for picking a fixed value for $K$ and looking at the curve $K=f(x,y)$. If a limit does not exist, that fact may be detected by going along one of the ``five paths": \\
\begin{enumerate}[i.]
\item $y$-axis
\item $x$-axis
\item $y=x$
\item $y=x^2$
\item $y=\sqrt{x}$
\end{enumerate}
\hrule
\begin{enumerate}[1.]
\item Let $f(x,y,z)=x^2y+xyz+z^2$. Compute $f(1,2,3)$. 
\item Consider the surface $z=4x-y$. Draw level curves of the surface corresponding to $K=-1, K=0$, and $K=1$. 
\item Consider the surface $z=\sqrt{9-x^2-y^2}$. Draw level curves of the surface corresponding to $K=0$, $K=2$, and $K=3$. 
\item Consider the surface $z=-2x^2-y$. Draw level curves of the surface corresponding to $K=-1$, $K=0$, and $K=1$. 
\item Show the limit does not exist: $\displaystyle\lim_{(x,y) \rightarrow (0,0)} \dfrac{xy}{x^2+3y^2}$. 
\item Show the limit does not exist: $\displaystyle\lim_{(x,y) \rightarrow (0,0)} \dfrac{3x^2+4y}{3x+4y}$. 
\item What is $\displaystyle\lim_{(x,y) \rightarrow (3,4)} e^{xy}(x^2-3y)$?
\item Let $f(x,y)= 3x^2y+5y^3$. Calculate $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$. 
\item Let $f(x,y,z)=e^{xy}(z^2x+z \ln(y)+x^2yz)$. Calculate $\dfrac{\partial f}{\partial x}, \dfrac{\partial f}{\partial y},$ and $\dfrac{\partial f}{\partial z}$. 
\end{enumerate}

\end{document}