Rate of Convergence:
Definition Let $\{x_n\}_{n\ge 0}$ is a sequence that converges to a number $x*$. Suppose that $\{a_n\}_{n\ge 0}$ is another sequence known to converge to $0$.
We say $\{x_n\}$ converges to $x*$ with rate of convergence of the sequence ${a_n}$ if there exists a positive constant $K$ such that
$$x_n - x* ≤ K \cdot a_n$$
for sufficiently large $n$. Normally we take $a_n = 1/n^p$. We are interested in the largest value $p$.
We deduce the following equation from above inequality $$ lim_{n\rightarrow\infty}\frac{x_n - x*}{a_n}= K$$
On what basis the inequality is converted to equality? In my book it has been done...
First, we should write $|x-x_n|\leq K |a_n|,$ not $x-x_n\leq K a_n.$ For if $x<x_n$ for every $n,$ then $x-x_n<0\leq K a_n$ for any $K>0$ and any non-negative sequence $a_n$, which tells us nothing about the rate of convergence of $x_n$.
Second, we cannot deduce that $(x_n-x)/a_n$ has a limit, and it usually doesn't. For example, let $x=x_n=1$ when $n$ is even, and let $x_n=1+1/(1+n)$ when $n$ is odd. Let $a_n=1/(1+n)$ for all $n$ and let $K=1.$ For even $n$ we have $(x_n-x)/a_n=0$, and for odd $n$ we have $(x_n-x)/a_n=1.$
When $x_n$ and $a_n$ are monotonic the ratio $|(x_n-x)/a_n| $ will still generally fail to converge.
Third, for rates of convergence, it doesn't matter what the values of $x_n$ are for any given finite collection of $n$. So it is equally useful if $|x-x_n|\leq K|a_n|$ for all but finitely many $n.$