I want to show that a function approaches 0 faster than other. The function is rational function and I want to show that numerator approaches faster towards 0. I want to prove it for the general case : if f(x)/g(x) =0 as x tends to "a" then how to prove that f(x) approaches 0 faster than other .
F(x)=g(x)=0 as x approaches "a"
For a polynomial, $f$, to approach zero as it approaches $a$, it must be zero at $a$ of some order. That means $f(x) = (x-a)^n p(x)$ where $p(x)$ is a polynomial that does not vanish at $a$. You can determine $p$ by factoring or long division. For instance $x^2-1$ vanishes at $1$, and we can write it as $(x-1) (x+1)$. $x+1$ is $p(x)$ in this case.
You can do the same with the denominator $g(x) = (x-a)^m q(x)$, where $q(x)$ is not zero at $a$.
Now you can express the rational function as $$\frac{f(x)}{g(x)} = (x-a)^{n-m} \cdot \frac{p(x)}{q(x)}.$$
$p(x)/q(x)$ converges to $p(a)/q(a)$ as $x\to a$. And if $n > m$ we have $(x-a)^{n-m} \to 0$ as $x \to a$. Thus $\lim_{x\to a} \frac{f(x)}{g(x)} = 0$ when $n > m$.