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{\bf HW 3\\Math 261, S19}

Please see the course syllabus for details on how to turn in your homework assignments.  This one is due at the beginning of your class on \underbar{\bf Friday, February 22}.

\begin{enumerate}

\item (5 pts.) \textsc{True or False}
\begin{enumerate}
\item Let $h(x,y)$ be a continuous function. Then for any poiny $(x_0, y_0)$ in the domain of $h$, the limit of $h$ as $(x,y)$ approach the point $(x_0, y_0)$  exists.
\item Let $h(x,y) = x/y$. The limit of $h$ as $(x,y)$ approach the point $(1,1)$  exists.
\item  Let $h(x,y) = x/y$. The limit of $h$ as $(x,y)$ approach the point $(0,0)$  exists.
\item  Let $h(x,y) = \left\{\begin{array}{cl}  3 & (x,y) = (0,0)\\ 1 & (x,y) \not= (0,0) \end{array} \right.$.  The limit of $h$ as $(x,y)$ approach the point $(0,0)$ doesn't exist.
\item   Let $h(x,y) = \left\{\begin{array}{cl}  3 & (x,y) = (0,0)\\ 1 & (x,y) \not= (0,0) \end{array} \right.$.  The limit of $h$ as $(x,y)$ approach the point $(0,0)$ is equal to 3.
\end{enumerate}

\item (3 pts.) If $f(x,y,z)=\sqrt{x^3+\sin(y) - y\ln(z)}$, find $f(2,\frac{\pi}{2},1)$. Perform elementary simplifications.

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\item (3 pts.)  Sketch the domain of $g(x,y) = \ln (1-2x-2y)$.

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\item  (3 pts.) Let { $\displaystyle h(x,y,z) = 3x^2z + z\cos(\pi y - \pi x) +  3e^z$}.  Determine $\lim_{(x,y,z)\to (1,2,0)} h(x,y,z)$.

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\item  (3 pts.) The function {  $ \displaystyle k(x,y) =  \frac{7x^8 y}{-2x^9+9y^9}$} has no limit as $(x,y) \to (0,0)$.\\ 

Show this by computing the limit of the function along the two following paths:
\begin{enumerate}
	\item  $t \mapsto (t,0)$. This notation indicates the path $(x(t),y(t)) = (t,0)$, or equivalently, the path given by $y=0$.
		\item $ t \mapsto (t,t)$. This notation indicates the path $(x(t),y(t)) = (t,t)$, or equivalently, the path given by $y=x$.
\end{enumerate}
{\bf Note (and hint):} the nice thing about the parametric  notation for the paths $t \mapsto (f(t), g(t))$ is that it suggests what you should do to compute the limit along the path: plug in the function $f(t)$ for $x$, the function $g(t)$ for $y$, and then take the limit as $t \to 0$. 

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\item  (3 pts.) Compute $\displaystyle \frac{\partial h}{\partial x}$ for the function in \#4.


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