# The Limit Constant Rule $ \lim_{x \to a} {[c f(x)]} = c \lim_{x \to a} f(x) $ ^2uw6gl ## Proof Given $\lim_{x \to a} f(x) = L$, by definition, for any $\epsilon$, there exists a range of $x$ values such that $|f(x) -L| \lt \epsilon$. $ \begin{gather} |f(x) -L| < \epsilon & \text{multiply both sides by $|c|$ where $c \ne 0$} \\ |c||f(x) -L | < |c| \epsilon \\ |cf(x)- cL| < |c|\epsilon & \text{pick $\epsilon = \frac {\epsilon’} {|c|}$} \\ |cf(x)- cL| < \epsilon’ & \text{reversing limit definition we can therefore state that} \\ \lim_{x\to a} [cf(x)] = cL \\ \lim_{x\to a} [cf(x)] = c \lim_{x \to a} f(x) \end{gather} $ One edge case remains, when $c = 0$. When $c = 0$ then $\lim_{x \to a} [c f(x)] = \lim_{x \to a} 0$, and also $c \lim_{x \to a} f(x) = 0 \cdot \lim_{x \to a} f(x) = 0$, so the rule stands for all $c$-s.