TITLE: If $f(x)\to f(a)$ when $x\to a$, why don't we denote it as $\displaystyle \lim_{x\to a}f(x)\to f(a)$?
QUESTION [6 upvotes]: If $f(x)\to f(a)$ when $x\to a$, why don't we denote it as
$$
\lim_{x\to a}f(x)\to f(a)
$$
instead of
$$
\lim_{x\to a}f(x) = f(a)?
$$
I need a comprehensible explanation for a newbie like me!

REPLY [2 votes]: Besides the conceptual reasons mentioned, it deserves emphasis that the reason for using an equation (vs. arrow relation) is that it allows us to use equational reasoning, e.g if $\:\,\rm\displaystyle \:\ell\, :=\, \lim_{x\to a}$
$$\rm \ell(f+gh)\ =\ \ell(f)\, +\, \ell(g)\ \ell(h)$$ 
which, assuming the limits exist and are known, may be evaluated by usual equational logic, here using the replacement rule, i.e. replacing equals by equals, i.e. using the equation $\rm\:\ell(f) = a\:$ to replace $\rm\:\ell(f)\:$ by $\rm\:a\:$ above, etc. In particular, if the functions are all polynomials this corresponds to evaluating the polynomials at $\rm\:x = a,\:$ i.e. using the equations $\rm\:f(x) = f(a)\:$ etc. to evaluate all of the polynomials at $\rm\:x = a.\:$ The algebraic (equational)  essence of the matter is clarified when one studies purely algebraic analogs in abstract algebra (congruences, evaluation homomorphisms, etc).  In fact, it is possible to interpret many seemingly analytic results purely algebraically, e.g. the product rule for limits can be viewed as a special case of the congruence product rule.
Generally, whenever one can convert relational properties to equational properties it yields simplificatons, e.g. converting congruence relations $\rm\:a\equiv b\,\ (mod\ m)\:$  to equations in the quotient ring $\rm\,\Bbb Z/m\, =\, $ integers mod $\rm\,m,\:$ which allows us to reuse our strong intuition manipulating integer equations on analogous modular equations.