Limit of Real Function of 2 Variables/Examples/x^2 + y^2 at 0

Example of Limit of Real Function of 2 Variables

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

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

Then:

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

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

Proof

Let $\epsilon$ be arbitrary.

Let $\delta = \sqrt {\epsilon / 2}$.

Then we have:

$\size x < \sqrt {\epsilon / 2}$

and:

$\size y < \sqrt {\epsilon / 2}$

together imply that:

$x^2 + y^2 < \epsilon$

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 $\text A$