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

Prove Limit 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

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