Linearity of the Limit Operator

Theorem: Linearity of the Limit Operator

If $f,g:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ have limits for $\mathbf{x}\to \mathbf{a}$, then for all $\lambda ,\mu \in \mathbb{R}$

$$\underset{\mathbf{x}\to \mathbf{a}}{\lim }[\lambda f(\mathbf{x})+\mu g(\mathbf{x})]=\lambda \underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x})+\mu \underset{\mathbf{x}\to \mathbf{a}}{\lim }g(\mathbf{x})$$

PROOF

TODO