What is the criteria for the Good approximation ?
e.g. we can approxiamte $\sin(x)$ to $x$ for $x<0.16 rad $ why 0.16 ? why not 0.23
enother e.g. $\tanh(x/2s)=x$ for $x<s$
and so on .. how to determaine the bounds , finding the max possible value for Good approximation , in other words : When to consider a Good Approxiamtion ?
On
It depends on the situation. Say for instance, we wanted to approximate $\sin(x)$ by $x$ with an error of less than $0.001$ - then we should solve $$|\sin(x)-x|<0.001$$ Which can be read as "The distance between $\sin(x)$ and $x$ is less than $0.001$". If we solve this (numerically), we get that this is true up until $x$ is about $0.18$ radians, and then the two start to get further apart.
More technically, when mathematicians use terms like "approximation" in proofs, they are usually concerned with how certain quantities behave in the limit, but I expect that isn't what you were asking.
Well, the criterion is for the inequality you want to be true... for instance, you known that $\sin x \operatorname*{\sim}_{x\to 0} x$ — and even more, $\sin x = x - \frac{x^3}{6} + o(x^4)$. This clearly means that on some neighborhood of zero, $[-u,u]$, you have $\sin x \in [x - \epsilon, x+\epsilon]$ for any $\epsilon>0$ of your choice (for instance). How to find a "good" $u$?
The most direct way is trying to solve the inequality by using all the tricks (convexity, etc) you have; or, if there is no elegant and quick way, to study for instance the function $\phi\colon x \to \sin x - x $ (and see when it is bounded by $\epsilon$) with all the tools of calculus.
You can also look for a mutiplicative approximation of the form $\sin x \in [(1 - \epsilon)x, (1+\epsilon)x]$. Then, considering $\psi\colon x \to \frac{\sin x}{x} $ and seeing when it's bounded by $(1\pm \epsilon)$ becomes the "generic, brute-force" approach.