About a week ago, I learned about linear approximation from a great youtube video, it was by Adrian Banner and the series of his lectures I think were from his book Calculus LifeSaver. I truly thought it was so beautiful and powerful a concept. Shortly I also got to know the Taylor Series and the general concept of this technique of matching the derivatives of any function with that of an approximating polynomial around a point. I messed with it a little and was so amazed by its success in predicting functions like sines and cosines and exponentials. For these functions, the polynomial approximation is true for any x. But for others, I found it not to be so. So, why is it that certain functions like e^x and trigonometric functions have so closely fitting Taylor approximations and why not the others? Does it have to do with the convergence of the approximation polynomial? Or is there more to it than just that?