Limits and Continuity With Given Constants

Question

I am given that:

$f(x)=cx$, where $c$ is a constant

Show that $\lim_{x→x_0}f(x)=f(x_0)$

I'm not really sure how to start this, any help would be greatly appreciated!

Answer 1

Let $\epsilon>0$. Take $\delta=\frac{\epsilon}{|c|+1}$. If $0<|x-x_0|<\delta$ then $$|f(x)-f(x_0)|=|cx-cx_0|=|c||x-x_0|<|c|\cdot\delta=|c|\cdot\frac{\epsilon}{|c|+1}<\epsilon.$$