Limit of Real Function of 2 Variables/Examples

Examples of Limits of Real Functions of 2 Variables

Limit of $x^2 + y^2$ at $0$

Let $f$ be the real function of $2$ variables defined as:

$\map f {x, y} = x^2 + y^2$

Then:

$\ds \lim_{\substack {x \mathop \to 0 \\ y \mathop \to 0} } \map f {x, y} = 0$

where $\lim$ denotes the limit of $f$.

Limit of $\dfrac {x - y} {x + y}$ at $0$

Let $f$ be the real function of $2$ variables defined as:

$\map f {x, y} = \begin {cases} \dfrac {x - y} {x + y} & : x \ne -y \\ 1 & : x = -y \end {cases}$

Then:

$\ds \lim_{\substack {x \mathop \to 0 \\ y \mathop \to 0} } \map f {x, y}$

does not exist.