Multivariable Functions

Functions that feature multiple independent variables.

Graphing

• Always one additional dimension for graphing.
  • \(f:D\subset\mathbb{R}^n\rightarrow\mathbb{R}^{n+1}\)

Limits and Continuity

\[
\lim_{(x,y) \to (0,0)}
\frac{xy^2}{x^2+y^4}
\\newline
let x = 0
\lim_{(x,y) \to (0,0)}
\frac{(0)y^2}{(0)^2+y^4} = 0
\\newline
let x = y^2
\lim_{(x,y) \to (0,0)}
\frac{(y^2)y^2}{(y^2)^2+y^4}
\frac{y^4}{2y^4} = 1/2
\\newline
1/2 \not= 0
\]