By the multiplication rule of limits, if a limit exists and is equal to a finite number, if we multiply our limit by a constant the result will be the limit value times the constant:
[tex]\begin{gathered} \lim_{x\to c}nf(x)=n\lim_{x\to c}f(x) \\ n\in\mathbb{R} \end{gathered}[/tex]Then, applying this property in our problem, we have:
[tex]\lim_{x\to c}cf(x)=c\lim_{x\to c}f(x)=c\cdot(L)=c\cdot L[/tex]