Derivative at a point (Linear approximation at point)--what is the valid range for approximation? Derivative at a point $x=c$ for $f(x)$ gives linear approximation (approaching a tangent) at that point.
But what is the range around $x=c$ for this approximation is valid? What books say is only $\lim_{\Delta\to 0}$?
Assume $f(x)$ = $x^2$. At $x=3$, I have derivative($f(x)$ = $2x$ of 6 which is actually a limit.
From right of x=3 i,e ${x \to 3^+}$,we should have   ${\Delta x\to 0}$ . 
Example 1 for  ${\Delta x\to 0}$ is 0.3,0.2,0.1,0.01,0.001,0.00001..........
Example 2 for  ${\Delta x\to 0}$ is 0.4,0.3,0.2,0.0,0.01,0.001,0.00001..........
I am looking for maximum first term that is allowed.
 A: An approximation can only be deemed good if you specify what good is. A quick demonstration: "A good approximation to $\pi$ is that it is roughly $7654.88$" is a statement that is as true as "A good approximation to $\pi$ is that it is roughly $3.14159265$". Both have no meaning before you specify 'good for what'. It will be correct to say that "$3.14159265$ is a better approximation (or at least not worse) to $\pi$ then $7654.88$ is, regardless of what you want the approximation for". 
Your question can be made precise and has many interesting and important answers. You will surely get to see Taylor approximations to functions and then you will learn about various forms for the remainder. I believe that will answer your question fully as you will see there conditions that assure that you can actually say something sensible about the linear approximation (and higher order approximations) of a given function. However, for a general differentiable function with no other knowledge other than differentiability there is nothing that can be said about the quality of the approximation. A famous function is Cauchy's function $f(x)e^{1/x^2}$ whose Taylor approximation is constantly $0$. 
