calculus - How do I find $limlimits_{x to 0} xcot(6x) $ without using L'hopitals rule?

How do I find $\lim\limits_{x \to 0} x\cot(6x) $ without using L'hopitals rule?

I'm a freshman in college in the 3rd week of a calculus 1 class. I know the $\lim\limits_{x \to 0} x\cot(6x) = \frac{1}{6}$ by looking at the graph, but I'm not sure how to get here without using L'hopital's rule.

Here is how I solved it (and got the wrong answer). Hopefully someone could tell me where I went wrong.

$\lim\limits_{x \to 0} x\cot(6x) = (\lim\limits_{x \to 0} x) (\lim\limits_{x \to 0}cot(6x)) = (0)((\lim\limits_{x \to 0}\frac{cos(6x)}{sin(6x)}) = (0)(\frac{1}{0}). $ Therefore the limit does not exist.

I am also unsure of how to solve $\lim\limits_{x \to 0} \frac{\sin(5x)}{7x^2} $ without using L'hopital's rule.