In-class work for 15 June 2012
1.) If the limit exists, compute $\displaystyle\lim_{(x,y) \rightarrow (0,1)} \frac{x^2+y^2}{xy+1}$. If the limit does not exist, why?

2.) If the limit exists, compute $\displaystyle\lim_{(x,y) \rightarrow (1,-1)} \log(x+y)$. If the limit does not exist, why?

3.) If the limit exists, compute $\displaystyle\lim_{(x,y) \rightarrow (\pi, \pi)} x \sin \left( \frac{x+y}{4} \right)$. If the limit does not exist, why?

4.) Use polar coordinates to compute the following limit: $\displaystyle\lim_{(x,y) \rightarrow (0,0)} \frac{e^{\sqrt{x^2+y^2}}-1}{\sqrt{x^2+y^2}}$.

5.) Notice that $\displaystyle\lim_{x \rightarrow 0} \left( \displaystyle\lim_{y \rightarrow 0} \frac{xy}{x^2+y^2} \right) = \displaystyle\lim_{x \rightarrow 0} \frac{0}{x^2} = 0$. Similarly, $\displaystyle\lim_{y \rightarrow 0} \left( \frac{xy}{x^2+y^2} \right) = \displaystyle\lim_{y \rightarrow 0} \frac{0}{x^2} = 0$. From these two facts, what can you conclude about $\displaystyle\lim_{(x,y) \rightarrow (0,0)} \frac{xy}{x^2+y^2}$? Why?

6.) Determine the set of points where the function is continuous: $f(x,y) = \frac{\sin(xy)}{e^x - y^2}$.

7.) Determine the set of points where the function is continuous: $g(x,y) = \arctan(x+\sqrt{y})$.

8.) If the limit exists, compute $\displaystyle\lim_{(x,y) \rightarrow (0,0)} e^{\frac{1}{xy}}$. If the limit does not exist, why?

9.) If the limit exists, compute $\displaystyle\lim_{(x,y) \rightarrow (0,0)} \frac{x^2}{x^2+y^2}$. If the limit does not exist, why?