Why can we replace a variable with a constant in a limit? We say $\lim_{x \to c} f(x) = L$ that means $f(x)$ may be as close to $L$ as $x$ tends to $c$. "Tends" here means $x$ approaches $c$ but never actually becomes $c$. If it so, then why do we such easily replace $x$ with value $c$, whenever it is appropriate. E.g. $\lim_{x \to 5} 4 + x = 4 + 5 = 9$, or $\lim_{h \to 0} f(x+h) = f(x)$.
Understand me right. I don't want to discuss cases when we can do the substitution and sometimes not, because we get a division by zero, and we need to do some simplification, etc. I understand all this. I just can't understand if $x$ actually is never $c$, what allows me to write $c$ as a value of $x$? Well, I used to think about it like "ah, as $x \to 0$, x is very small number, let it be zero". But it is not statistics, you know, to close eyes and make approximations.
I met the notion of "infinitesimal" and as I understood it is opposed to "$\delta-\epsilon$" approach. I can't fully understand how they are related to each other and to my question. Maybe I lack of historical context. If it so, please clarify this for me. Thanks.

In order to compute $\lim_{x \to 5} (4 + x)$, we use the fact that
$$\lim_{x \to c} (f(x) + g(x)) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$$
whenever both limits on the right exist.
Now, $\lim_{x \to 5} 4 = 4$ holds because as $x$ approaches any value (including $5$), the expression $4$ approaches the value $4$, since it is constant.
Also, $\lim_{x \to 5} x = 5$ because as $x$ approaches $5$, the expression $x$ approaches $5$, trivially.
Of course, all of these results can be rigorously proven with the $(\epsilon, \delta)$-definition of limit.
Thus $\lim_{x \to 5}(4 + x) = 4 + 5 = 9$. It looks as though we just replaced $x$ with $5$, but what is actually going on is quite different.
