Evaluate the limit: $\underset{x\to 0}{\lim }\left(\mathrm{cosec}x-\cot x\right)$

Let $a_1, a_2, \dots , a_n$ be fixed real numbers and define a function 
$f\left(x\right)=\left(x-a_1\right)\left(x-a_2\right)\dots \left(x-a_n\right)$

What is $\underset{x\to a_1}{\lim }f\left(x\right)$? For some $a\ne a_1, a_2, \dots , a_n$, compute $\underset{x\to a}{\lim }f\left(x\right)$.