Prove that $\lim\limits_{x→4}x^{1/2}=2$ by using epsilon-delta

Question

Prove that $\lim\limits_{x→4} x^{1/2}=2$ given $ε=1$ by using epsilon-delta.

I know that given $ε>0$, $∃ δ>0$ such that if $0<|x−4|<δ$, then $|x^{1/2}-2|<ε$. The thing is I am stuck on how to expand the previous expression to find $δ$.

Can anyone help me? Thank you.

Answer 1

Guide:

$$\left|\sqrt{x} - 2 \right|=\left|\frac{x-4}{\sqrt{x}+2} \right| $$

• Find a lower bound for $\sqrt{x}+2$, you can impose conditions such as $\delta$ can't be bigger than $1$.

Answer 2

You can work the problem like this: $$|x^{1/2}-2| < |x^{1/2}+2||x^{1/2}-2| = |x-4|$$ So if you take $\delta = \epsilon$, we are done.