It is usual in physics, that when we have a variable that is very small or very large we do a power series expansion of the function of that variable, and eliminate the high order terms, but my question is, why do we usually make the expansion and then approximate, why don't we just do the limit in that function, when that value is very small (tends to zero) or is very large (tends to infinity).
[Physics] The reasoning behind doing series expansions and approximating functions in physics
approximationsmathematics
Best Answer
The key reason is that we want to understand the behavior of the system in the neighborhood of the state rather than at the state itself.
Take the equation of motion for a simple pendulum, for example:
$$\ddot{\theta} = -\frac{g}{\ell}\sin(\theta)$$
If we take the limit where $\theta \rightarrow 0$, we find $\ddot{\theta}= 0$, and we would conclude that the pendulum angle increases or decreases linearly with respect to time.
If we however take a Taylor expansion and truncate at the linear term, we find $\ddot{\theta} = -\frac{g}{\ell}\theta$, which is a simple harmonic oscillator! This expansion shows us that in the neighborhood of $0$, the system returns back to $0$ as if it was a simple harmonic oscillator: completely unlike what we could state in the limit approximation above.
In fact, you could consider the limiting behavior around a state to be the zeroth-order component of a local expansion, which holds true straightforwardly for the example above since the limit term contributes no terms to the dynamics of the pendulum (but correctly notes that the angle increases/decreases linearly very close to $0$).