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

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!

Answer 1

$\displaystyle \lim_{x \to a} f(x)$ denotes a value which is constant, so it cannot it any meaningful and nontrivial sense tend to anything. However, $f(x)$ refers to a variable quantity, depending on the value of $x$, so a notion of tending to has a meaningful and nontrivial meaning.

Answer 2

This would be because $\lim_{x\to a} f(x)$ is $f(a)$.

Answer 3

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.

Answer 4

The notation $\lim_{x\to a} f(x)$ stands for the number $z$ satisfying the following $ \forall \epsilon >0 : \exists \delta: |x-a|<\delta: |f(x)-z|<\epsilon, $ if this $z$ exists.

So $\lim_{x\to a} f(x)$ stands for a number and thus cannot change as the arrow you suggests implies. The fact that $\lim_{x\to a} f(x)=f(a)$ is used means that this number $z$ is f(a).

Note that there can only be one such number $z$ and this number does not necesarily exist. This is off course the difference between continuity and non continuity.