I'm reading a real Analysis book in which Taylor's Theorem was introduced in chapter 10 and now the next two chapters are 11. Maxima and Mamina and 12. Indeterminate Forms. In both the chapters the author says that this chapter is an application of Taylor's Theorem but when I read the theory portion I didn't find anything related to Taylor's Theorem ( Maybe this book doesn't explain the relation ) So my question is
- How is Maxima and Mamina an application of Taylor's Theorem ( i.e how is it related to it)
- Same as 1) query for Indeterminate Form
Edit : After thorough reading I did see series expansion in Theorem and Functions for Max/Min condition and while evaluating limit respectively Anything else that's major that I'm missing?
This is because Taylor's Theorem is just an approximation of the local behaviour of the function around a point. Therefore, given a point $x_0$ and a function $f(x)$, we can use Taylor's expansion of $f(x)$ around $x_0$ and, by analysing its behaviour, we can understand if $x_0$ is a local minima or maxima for $f(x)$ (that is if the first derivative $f'(x_0) = 0$ and if the second derivative $f''(x_0) > 0$ or $f''(x_0) < 0$).
The reason why it can be seen as an application of Taylor's Theorem is because indeterminate forms can be resolved easily thanks to it. More in particular, we can approximate both functions at the limit point using Taylor's expansion, and we can analyse how they behave one with respect to another more closely. For instance, let's have: $$ \lim_{x\to x_0}\frac{f(x)}{g(x)} = \frac{0}{0} $$ We can expand both $f(x),g(x)$ and we get a polynomial expression. Then, depending on how fast the functions tend to $0$, we get the result.