Question

Prove the statement using the ε, δ definition of a limit. lim x → 1 4 + 2x 3 = 2

Answer 1

You want to prove

$\displaystyle\lim_{x\to1}\frac{4+2x}3=2$

By the definition of the limit, this means that for any $\varepsilon>0$, we can find $\delta>0$ such that

$|x-1|$.

In order to get the inequality we want, we need to have

$\left|\dfrac{4+2x}3-2\right|=\left|\dfrac{2x-2}3\right|=\dfrac23|x-1|$

so we can pick $\delta=\frac{3\varepsilon}2$.

So here's the proof: Let $\varepsilon>0$ be given, and let $\delta=\frac{3\varepsilon}2$. Then

$|x-1|$

$\implies\dfrac23|x-1|$

$\implies\left|\dfrac{2x-2}3\right|=\left|\dfrac{2x+4}3-1\right|$

as required.