Lecture 010

Limits and Continuity

Limit in 3D: if $0 \leq \text{distance}((x, y), (a, b)) < \delta \implies 0 \leq |f(x, y)-L| < \epsilon$ the limit of $z = f(x, y)$ is $\lim_{(x, y) \rightarrow (a, b)} f(x, y) = L$

Find the Limit when Exists

1. try to plug in
2. try to simplify, cancel denominators, plug in
3. If get $\frac{0}{0}$, we need to show $\lim_{(x, y) \rightarrow (a, b)} f(x, y)$ does not exists
4. $\frac{\text{something}}{0}$ means derivative DNE

Prove Limit DNE:

1. Write arbitrary $x, y, z, ...$ all in terms of polynomials of one variable, say $x$.
2. Find some polynomials $X_1, X_2, Y_1, Y_2$ such that $\lim_{(X_1, Y_2, ...) \rightarrow (a, b, ...)} f(x, y, ...) \downarrow \neq \lim_{(X_2, Y_2, ...) \rightarrow (a, b, ...)} f(x, y, ...) \downarrow$ and none of them are $\frac{0}{0} \uparrow$