Exercise 18

For each of the following prompts, provide an example of a function of two variables with the desired properties (with justification), or explain why such a function does not exist.

1. A function $p$ that is defined at $(0,0)$, but $\lim_{(x,y) \to (0,0)} p(x,y)$ does not exist.

2. A function $q$ that does not have a limit at $(0,0)$, but that has the same limiting value along any line $y = mx$ as $x \to 0$.

3. A function $r$ that is continuous at $(0,0)$, but $\lim_{(x,y) \to (0,0)} r(x,y)$ does not exist.

4. A function $s$ such that $$\begin{gathered} \lim_{(x,x) \to (0,0)} s(x,x) = 3 \ \ \ \mbox{and} \ \ \ \lim_{(x,2x) \to (0,0)} s(x,2x) = 6, \end{gathered}$$ for which $\lim_{(x,y) \to (0,0)} s(x,y)$ exists.

5. A function $t$ that is not defined at $(1,1)$ but $\lim_{(x,y) \to (1,1)} t(x,y)$ does exist.