Limit of Real Function of 2 Variables/Examples/(x - y) over (x + y) at 0

Example of Limit of Real Function of 2 Variables
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.

However, note that:

Proof
Let $\epsilon$ be arbitrary.

Consider $\tuple {x, y}$ along the $L$ defined as $x = -y$.

Then selecting points close to $L$ we can make $\dfrac {x - y} {x + y}$ as large as we like by making $x - \paren {-y}$ as small as we like.

Hence any square around $\tuple {0, 0}$ is going to contain $\tuple {x, y}$ for which $\map f {x, y}$ is greater than any $\epsilon$ you pick.

Hence the result.