I got to read about Linearization and Quadratic approximation and in general approximation theory. From what I observed from the examples discussed there, it seems like the approximation works only for points closer to the known point x = a. For points farther away from that point x=a, we may get more error in the approximation so it is not preferable.

**Question 1:**

Is my understanding about region of approximation correct ?

**Question 2:**

If we extend the degree of the approximation terms in taylor series to reasonable n, what is exactly happening ? how does the accuracy increase ? what if there are many bumps in curve near the point of approximation ? Can someone give an example for a complex function and show ?

**Question 3:**

If the above approximation method just approximates the region surrounding a point, i would like to know if there is any way to determine a approximation of the entire function (not just region close to desired point) and get a decent error ? if so what is it ?

Is this technique (answer of above part in question 3) is the one used in machine learning(in computer science) in regression to predict the new output when we already know a set of old inputs and outputs ?