limit of a (system) function

Question

Sorry if my title doesn't explain very well. (I don't know how should I translate it). But I have this problem:
I had this function: $$ f(x) = \begin{cases} 1 & \text{$x$ $\neq$ 0} \\ 0 & \text{$x$ = 0} \end{cases} $$
Can anyone help me why $$\lim_{x\to0} f(x) = 1$$ and not 0?

Answer 1

$$\lim_{x\to0,x\in\mathbb R^*}f (x)=1$$ because $$(\forall x\in\mathbb R^*)\;\;f (x)=1$$ and we write $$\lim_{x\to0,x\ne0}f (x)=1$$ but $$\lim_{x\to0,x\in\mathbb R}f (x) $$ doesn't exist.

Answer 2

Simply because, when calculating the limit of a function as $x$ tends to a certain value $V$, the point where $x=V$ is not included. Since $f(x)=1$ everywhere except $x=0$, and here $V=0$, this just means that the limit must be $1$.