lim cos(1/θ) = 0, when θ → 0. Why? Let $f(x)=x^2\sin\left(\frac{1}{x}\right)$ for $x\ne0$ and $f(0)=0$. =>
If we use Lagrange's theorem: $\exists \theta \in (0;x)$ and $f(x) - f(0) = f′(\theta)(x-0)$ =>
$$x^2\sin\left(\frac{1}{x}\right) - 0 = \left(2x\sin\left(\frac{1}{\theta}\right) - \cos\left(\frac{1}{\theta}\right)\right)(x - 0)$$ =>
Because $x>0$, we can divide both parts of equality on it =>
$$x\sin\left(\frac{1}{x}\right) = 2x\sin\left(\frac{1}{\theta}\right) - \cos\left(\frac{1}{\theta}\right)$$
Also notice: if $x \to 0$, than obviously $\theta \to 0$ =>
$$\lim_{x\to 0}\sin\left(\frac{1}{x}\right) = \lim_{x\to 0}2x\sin\left(\frac{1}{\theta}\right) - \cos\left(\frac{1}{\theta}\right)$$ =>
$$0 = 0 - \lim_{x\to 0}\cos\left(\frac{1}{\theta}\right)$$ =>
$$\lim_{\theta \to 0}\cos\left(\frac{1}{\theta}\right) = 0$$
Why?
 A: The issue is subtle, it is exactly the reason why the converse of L'Hospital rule doesn't work.
For each $x$ you get an $\theta_x$ with that property.
All you prove is that $0< \theta_x < x$ and 
$$\lim_{x \to 0}\; \cos\left(\frac{1}{\theta_x}\right)=0.$$
And yes, $x \to 0$ implies that $\theta_x \to 0$, but $\theta_x$ is not a free variable converging to $0$. It is possible for example that all $\theta_x$ have the form $\frac{2}{(2n_x+1)\pi}$ where $n_x$ is an integer depending on $n$.
So the issue is that, even if $\theta_x \to 0$, you cannnot infer that 
$$\lim_{x \to 0}\; \cos\left(\frac{1}{\theta_x}\right) = \lim_{\theta \to 0}\; \cos\left(\frac{1}{\theta}\right).$$
This equality is actually false.
A: As $\theta$ approaches $0$ from the right, $\frac{1}{\theta}$ approaches infinity. It's easy to see that cosine does not have a limit as its input approaches infinity because the output reliably vibrates between $1$ and $-1$ forever. It does not settle down to any fixed value.
By the way, thanks for including your work. It shows us where numerous mistakes were made. Among these you dropped some terms, incorrectly evaluated a limit of $\sin(1/x)$ and fallaciously reasoned that you could change some but not all of the variables from terms of $\theta$ into terms of $x$ and remove some that way. Try not to repeat these mistakes!
