Question

prove the statement using the epsilon, delta definition of a limit. lim of x to 1 to (2+4x)/3=2

Answer 1

\[\text{let }\epsilon>0\]

\[\text{choose }\delta=\frac{3}{4}\epsilon\]

then

\[\left|\frac{4x+2}{3}-2\right|=\left|\frac{4x+2-6}{3}\right|=\left|\frac{4x-4}{3}\right|=\frac{4}{3}\left|x-1\right|<\frac{4}{3}\delta=\frac{4}{3}\frac{3}{4}\epsilon=\epsilon\]