Section 6.1: 1, 3, 4, 7, 11, 13, 19, 21, 22, 23

Also:

1. Let $\displaystyle{f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}}$.
   1. Show if $(x, y) = (t, 0)$, then $\displaystyle{\lim_{t \to 0}f(x, y) = 1}$.
   2. Show if $(x, y) = (0, t)$, then $\displaystyle{\lim_{t \to 0}f(x, y) = -1}$.
   3. Show if $(x, y) = (t, t)$, then $\displaystyle{\lim_{t \to 0}f(x, y) = 0}$.
   4. Does $\displaystyle{\lim_{(x, y) \to (0, 0)}f(x, y)}$ exist? Explain.
2. Let $\displaystyle{f(x, y) = \frac{x^2y}{x^4 + 4y^2}}$.
   1. Show if $(x, y) = (t, 0)$, then $\displaystyle{\lim_{t \to 0}f(x, y) = 0}$.
   2. Show if $(x, y) = (0, t)$, then $\displaystyle{\lim_{t \to 0}f(x, y) = 0}$.
   3. Show if $(x, y) = (t, t)$, then $\displaystyle{\lim_{t \to 0}f(x, y) = 0}$.
   4. Show that for any scalar $m$ if $(x, y) = (t, mt)$, then $\displaystyle{\lim_{t \to 0}f(x, y) = 0}$.
   5. Show if $(x, y) = (t, t^2)$, then $\displaystyle{\lim_{t \to 0}f(x, y) = \frac{1}{5}}$.
   6. Does $\displaystyle{\lim_{(x, y) \to (0, 0)}f(x, y)}$ exist? Explain.