Can the limits and derivatives of a function always be solved by applying its basic theorems? I'm studying calculus, specifically about the limits and derivatives of a function. So I became interested in studying the basic theorems and the properties of limits and derivatives of a function. However, throughout the theory, the following question arises:
Can the limits and derivatives of a function always be solved by applying its basic theorems?
I would like to know if this is true or false and I think the best way to do it is through an example, and I am looking for it.

No. There are another theorems which are necessary for indeterminate limits.
For example the classical limit
$$ \lim_{h\to 0} \frac{\sin(h)}{h}.$$
It can't be solved using only the basic theorems. In your course you will learn a specific proof that demostrates that this limit equals 1.
