Vtyjkyuk

By doubt is, if we prove the product rule of limits,can't we directly prove the quotient rule ? But wherever I look on the internet, they give another $epsilon-delta$ proof for the quotient rule, why is that required ?
My proof of the quotient rule :

$lim_xto afracf(x)g(x)$ = $lim_xto af(x)$$lim_xto afrac1g(x)$ . (Accepting the product rule)

= $fraclim_xto af(x)lim_xto ag(x)$ $lim_xto afrac1g(x)$$lim_xto ag(x)$ Multiplying and dividing by $lim_xto ag(x)$

= $fraclim_xto af(x)lim_xto ag(x)$ $lim_xto a[frac1g(x)g(x) ]$ (Accepting the product rule)

= $fraclim_xto af(x)lim_xto ag(x)$ $lim_xto a1$

= $fraclim_xto af(x)lim_xto ag(x)(1)$ (Accepting $lim_xto ac$ = c )

And hence the result. Am I missing something here, maybe this is circular reasoning? Why does this not work ?

You are only given that
$$ lim_xto a g(x) = B not=0. $$
Without the quotient rule, you don't even know whether
$$ lim_xto a frac1g(x) $$
exists (or that it is equal to $1/B$), so that the use of the product rule in your first step is unjustified.

However, if you prove the statement above, then you can use the product rule to prove the general quotient rule
$$ lim_xto a fracf(x)g(x) = fracAB, $$
as you proposed.

EDIT: Just to be fully clear, here are the statements of the product rule and the quotient rule.

Suppose that $$lim_xto a f(x) = A, quad textand quad lim_xto a g(x) = B.$$
Then
$$ lim_xto a f(x) cdot g(x) = Acdot B. $$
If we also have
$$ lim_xto a g(x) = B not= 0, $$
then
$$ lim_xto a fracf(x)g(x) = fracAB. $$

Notice that it is nowhere assumed that
$$ lim_xto a frac1g(x) $$
exists. This needs to be proven (which basically amounts to proving the quotient rule).