# [SOLVED]Check existence of limit with definition

#### mathmari

##### Well-known member
MHB Site Helper
Hey!!

I want to check the existence of the limit $\lim_{x\to 0}\frac{x}{x}$ using the definition.

For that do we use the epsilon delta definition?

If yes, I have done the following:

Let $\epsilon>0$. We want to show that there is a $\delta>0$ s.t. if $0<|x-0|<\delta$ then $|f(x)-1|<\epsilon$.

We have that $\left |f(x)-1\right |=\left |\frac{x}{x}-1\right |=\left |\frac{x-1}{x}\right |=\frac{|x-1|}{|x|}$.

How can we continue?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Hey mathmari !!

Since $0<|x-0|$, it follows that $x\ne 0$.
Therefore $\left|\frac xx -1\right|=0$, isn't it?

#### mathmari

##### Well-known member
MHB Site Helper
Since $0<|x-0|$, it follows that $x\ne 0$.
Therefore $\left|\frac xx -1\right|=0$, isn't it?
Ohh yes! Thank you!!