By the definitions of cosecant, secant, and cotangent, we have
[tex]\dfrac{\sin2x\csc3x\sec2x}{x^2\cot^24x}=\dfrac{\sin2x\sin^24x}{x^2\sin3x\cos2x\cos^24x}[/tex]
Then we rewrite the fraction as
[tex]\dfrac{\sin2x}{2x}\left(\dfrac{\sin4x}{4x}\right)^2\dfrac{3x}{\sin3x}\dfrac{32}{3\cos2x\cos^24x}[/tex]
The reason for this is that we want to apply the well-known limit,
[tex]\displaystyle\lim_{x\to0}\frac{\sin ax}{ax}=\lim_{x\to0}\frac{ax}{\sin ax}=1[/tex]
for [tex]a\neq0[/tex]. So when we take the limit, we have
[tex]\displaystyle\lim_{x\to0}\cdots=\lim_{x\to0}\frac{\sin2x}{2x}\left(\lim_{x\to0}\frac{\sin4x}{4x}\right)^2\lim_{x\to0}\frac{3x}{\sin3x}\lim_{x\to0}\frac{32}3\cos2x\cos^24x}[/tex]
[tex]=1\cdot1^2\cdot1\cdot\dfrac{32}3=\dfrac{32}3[/tex]
