Find limit when $\theta$ tends to $0$ of $\tan(\theta) /\theta$ I understand that $\lim_{\theta\to0}(\sin(θ)/θ) = 1$ but what is $x$ when, 
$\lim_{\theta\to0}(\tan(θ)/θ) = x$ where $x$ is a real constant value. 

Please help me, I will be eternally great full :D
We have that $\dfrac{\tan\theta}{\theta}=\dfrac{1}{\cos\theta}\cdot \dfrac {\sin\theta}{\theta}.$ 

Then recall that the limit of a product is equal to the product of the limits (when those limits do in fact exist.) 

$$\lim_{\theta\to 0} \dfrac{\tan\theta}{\theta}= \lim_{\theta\to 0} \dfrac{\sin\theta}{(\cos\theta)\cdot \theta}=\lim_{\theta \to 0} \dfrac{1}{\cos\theta}\cdot \dfrac {\sin\theta}{\theta} = \lim_{\theta \to 0} \dfrac 1{\cos \theta} \cdot \lim_{\theta\to 0} \dfrac{\sin \theta}{\theta} = 1\cdot 1 = 1.$$
$x=1$

Let $\theta=y$. Then $$\tan y=\frac{\sin y}{\cos y}$$ 
and $\lim_{y \to 0} \frac{\sin y}{y}=1$ and $\cos 0=1$ so $\lim_{y\to 0}\frac{\tan y}{y}=0$.