Find $\displaystyle \lim_{x \to 0} \frac{x \cdot \operatorname{cosec}(2x)}{\cos(5x)}$ I am having difficulties to find the limit for
$$\lim_{x \to 0} \frac{x \cdot \operatorname{cosec}(2x)}{\cos(5x)}$$
I tried to get rid of $ \operatorname{cosec} $ fist
$$\lim_{x \to 0} \frac{\dfrac{x}{\sin(2x)}}{\cos(5x)}$$
Probably I should get it to a point where I could make use of $\displaystyle \lim_{x \to 0} \frac{\sin(x)}{x} = 1$ but I don't know how to continue.
Maybe if one could give me a hint for the next step?

The Taylor expansion of Csc[y] built around y=0 is 1/y + y/6. You may forget the denominator and arrive to x (1/(2x) + (x/3)) = 1/2
