How can I prove that I can assume $\sin{x}, \tan{x}$ to be x when $x \rightarrow 0$

Question

I know the proof of $\lim_{x\rightarrow 0}{\frac{\sin{x}}{x}}=1$ and $\lim_{x\rightarrow 0}{\frac{\tan{x}}{x}}=1$.

However, I do not think that this means that I can change every $\sin{x}$ and $\tan{x}$ inside a limit to x and solve the problem, because the rule of limits can only be apllied if both limits exist, and it also does not mention anything about composite functions (for example $\sin{(\sin{(x)})}$)

Is it possible to prove that every $\sin(x)$ and $\tan(x)$ can be canged into x no matter its location if $x\rightarrow 0$ Without Taylor Expansion

Answer 1

It can't be proven, because it's not true. For example: $$\lim_{x \to 0} \frac{x-\sin(x)}{x^3}=\frac{1}{6}$$ While $$\lim_{x \to 0} \frac{x-x}{x^3}=\lim_{x \to 0} \frac{0}{x^3}=0$$

Answer 2

See Taylor expansion of these functions : you can say that near $0$, they "look like" a polynomial and control the difference.

For example, $\sin x = x + x\cdot \varepsilon(x)$ where $\varepsilon$ is a continuous function of $x$ vanishing at $0$.