Why does $(x \to a) ≠ (x = a)$ , But $f(x \to a) = f(x = a)$ I am really satisfied that $(x \to a) ≠(x=a)$ and if that is not right , Then all the process of $Limits$ is dividing by zero and that is a crime.
Since $(x \to a) + h = (x=a)$ , $h ≠ 0$,So Why does $f(x \to a) = f(x = a)$ ?
NOTE:I am talking about continuous function.
 A: Your question seems unclear, but perhaps emphasizing this distinction will help:
A function $f: \mathbb R \longrightarrow \mathbb R$ is continuous at $a \in \mathbb R$ if (and only if) for every $\varepsilon>0$ there exists $\delta>0$ such that $|f(x)-f(a)|<\varepsilon$ whenever $|x-a|<\delta$.
We write $\lim_{x \to a} f(x)=L$ if and only if for every $\varepsilon>0$ there exists $\delta>0$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-a|<\delta$.
Noticing the differences between the two formal definitions, we see that a function $f$ is continuous at $a \in \mathbb R$ if and only if $\lim_{x \to a} f(x)=f(a)$.
A: Intuitively speaking, the number $$A=\lim_{x\to a} f(x)$$ is the value that the function $f$ “should” have at the point $a$, judging by the values $f(x)$ at the surrounding points ($x \neq a$).
And $$B=f(a)$$ is of course the value that the function $f$ does have at the point $a$.
These two numbers $A$ and $B$ need not be equal, but if they are equal, then the function $f$ is said to be continuous at the point $a$.
So when you say that you are talking about continuous functions, that's just another way of saying that you are only looking at function which happen to have this particular propery that $A=B$, i.e., $$\lim_{x\to a} f(x) = f(a)$$
(or “$f(x \to a)=f(x=a)$” in your notation, which I would not recommend).
