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:

$\ds \lim_{x \mathop \to 0} \paren {\lim_{y \mathop \to 0} \map f {x, y} }$ equals $1$

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

$\ds \lim_{y \mathop \to 0} \paren {\lim_{x \mathop \to 0} \map f {x, y} }$ equals $-1$

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

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.

$\blacksquare$

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 3$. Functions of Several Variables: $3.1$ Limits and Continuity: Example $\text B$