Solve limit without use L'Hopital: $\lim _{x\to 0}\left(\frac{sin\left(2x\right)-2sin\left(x\right)}{x\cdot \:arctg^2x}\right)$

Question

Task: $$\lim _{x\to 0}\left(\frac{\sin\left(2x\right)-2\sin\left(x\right)}{x\cdot \:arctan^2x}\right)$$

Hello! It is necessary to solve the next limit. I try several methods. For example, if you use the equivalent of small, it turns out: $$\sin\left(2x\right)\rightarrow 2x$$ $$2\sin\left(x\right)\rightarrow 2x$$ $$x\cdot arc\tan^2\left(x\right)\rightarrow x\cdot x^2$$ Limit obtained indefinite again...(

I have tried to simplify the numerator (only in the numerator as the denominator have no idea what to do): $$\lim _{x\to 0}\left(\frac{2\sin x\cdot \cos x-2\sin x}{x\cdot arc\tan^2x}\right)=\lim _{x\to 0}\left(\frac{\sin x\left(\cos x-2\right)}{x\cdot arc\tan^2x}\right)=?$$

Answer 1

Use equivalents:

Numerator: $\;\sin 2x-2\sin x=2\sin x(\cos x-1)$, and $\sin x\sim_0 x$, $\;\cos x-1\sim_0 -\dfrac{x^2}2$.

Denominator: $\;\arctan x\sim_0 x$.

Thus: $$\frac{\sin 2x-2\sin x}{x\arctan^2 x}\sim_0\frac{x(-x^2)}{x\cdot x^2}=-1.$$

Answer 2

HINT:

$$\dfrac{\sin2x-2\sin x}{x\arctan^2x}=-2\cdot\dfrac{\sin x}x\cdot\dfrac{1-\cos x}{x^2}\cdot\left(\dfrac x{\arctan x}\right)^2$$

and $$\dfrac{1-\cos x}{x^2}=\dfrac1{(1+\cos x)}\cdot\left(\dfrac{\sin x}x\right)^2$$

Answer 3

So you have the following limit:

$$\lim _{x\to 0}\left(\frac{\sin\left(2x\right)-2\sin\left(x\right)}{x\cdot \:\arctan^2x}\right)$$

I would suggest using taylor series:

$$\lim _{x\to 0}\left(\frac{\sin\left(2x\right)-2\sin\left(x\right)}{x\cdot \:\arctan^2x}\right) = \frac{\left(2x - \frac{(2x)^3}{3!} + \frac{(2x)^5}{5!} - \ldots\right) - 2\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\right)}{x\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\right)^2} = \frac{-x^3 + \ldots}{x^3 + \ldots}$$

Note that we only care about the lowest power of $x$ on the numerator and the denominator. In this case it is $x^3$. Thus, if we multiply the fraction by $\frac{1}{x^3}$, we end up with:

$$\frac{-x^3 + \ldots}{x^3 + \ldots} = \frac{-1 + 0 + \ldots}{1 + 0 + \ldots} = -1$$

Therefore our answer is $-1$. Comment if you have questions.