$\sin(x)$ is asymptotically equal to $x+5x^3$

Question

Here is my question: I've never seen before this kind of fact underlined about asymptotic equalities (and why we keep only one term in these equalities) and I'm looking for reference. Here is an example: we know that $$\sin(x) \underset{x\to 0}{\sim} x,$$ and also (the $5x^3$ is in fact arbitrary) $$ x+5x^3 \underset{x\to 0}{\sim} x,$$ so, since $\underset{x\to 0}{\sim}$ is an equivalence relation, $$\sin(x) \underset{x\to 0}{\sim} x+5x^3.$$ I've never seen this in a book or a course, do you have any reference for this amusing fact? Or others exercises on this topic?

Answer 1

We can reformulate the question like this.

$\sin(x) \underset{x\to 0}{\sim} x,$ means that $\lim_{x \to 0} \frac{\sin x}{x} = 1 $.

$ x+5x^3 \underset{x\to 0}{\sim} x,$ means that $\lim_{x \to 0} \frac{x+5x^3}{x}=\lim_{x \to 0}\left(1+5x^2 \right)=1$.

So since $=$ is also an equivalence relation, we can say that

$$\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{x+5x^3}{x} = 1. $$

Of course the rate of convergence could be different. To describe the behavior of the limits we can use asymptotic analysis.