# Thread: Proving statement using the symbol definition of a limit

1. ## Proving statement using the symbol definition of a limit

Prove that lim,x-->0 (x^2)=0 using ε,ζ.

-I am unsure how to approach this.

2. ## Re: Proving statement using the symbol definition of a limit

Originally Posted by johnsy123
Prove that lim,x-->0 (x^2)=0 using ε,ζ.

-I am unsure how to approach this.
To show that \displaystyle \begin{align*} \lim_{x \to 0}x^2 = 0 \end{align*} you need to show \displaystyle \begin{align*} |x - 0| < \delta \implies \left|x^2 - 0\right| < \epsilon \end{align*}.

\displaystyle \begin{align*} \left|x^2 - 0\right| &< \epsilon \\ \left|x^2\right| &< \epsilon \\ |x|^2 &< \epsilon \\ |x| &< \sqrt{\epsilon} \end{align*}

So if we choose \displaystyle \begin{align*} \delta = \sqrt{\epsilon} \end{align*} we will be able to show this implication...

\displaystyle \begin{align*} |x - 0| &< \delta \\ |x| &< \sqrt{\epsilon} \\ |x|^2 &< \epsilon \\ \left|x^2\right| &< \epsilon \\ \left|x^2 - 0\right| &< \epsilon \end{align*}

Therefore \displaystyle \begin{align*} |x - 0| < \delta \implies \left|x^2 - 0\right| < \epsilon \end{align*}.

Q.E.D.

3. ## Re: Proving statement using the symbol definition of a limit

Originally Posted by johnsy123
Prove that lim,x-->0 (x^2)=0 using ε,ζ.
This is a clearly easy question. Look $0 < \varsigma < 1 \Rightarrow 0 < {\varsigma ^2} < \varsigma < 1$
So let $\varepsilon = \varsigma~!$