Evaluate the following limit: $\underset{x\to \frac{\mathrm{\pi }}{6}}{\lim }\frac{\left(2{\sin }^{2}x+\sin x-1\right)}{\left(2{\sin }^{2}x-3\sin x+1\right)}$.

Evaluate the following limit: $\underset{x\to 0}{\lim }\frac{(\sin 2x+3x)}{(2x+\sin 3x)}$.

Evaluate the following limit: $\underset{x\to 0}{\lim }\frac{(\tan 2x-x)}{(3x-\tan x)}$.

Evaluate the following limit: $\underset{x\to 0}{\lim }\frac{\left(x^2-\tan 2x\right)}{\tan x}$.

Evaluate the following limit: $\underset{x\to 0}{\lim }\frac{(x\cos x+\sin x)}{\left(x^2+\tan x\right)}$.

Evaluate the following limit: $\underset{x\to 0}{\lim }\frac{(\tan x-\sin x)}{{\sin }^{3}x}$.

Evaluate the following limit: $\underset{x\to 0}{\lim }xcosecx$.

Evaluate the following limit: $\underset{x\to 0}{\lim }\left(x\cot 2x\right)$.

Evaluate the following limit: $\underset{x\to 0}{\lim }\frac{\sin x \cos x}{3x}$.