Question

Evaluate

${lim}_{x\to 0}\frac{cos\left(2x^3\right)-cos\left(5x^3\right)}{xsi{n}^2\left(2x\right)ta{n}^3\left(3x\right)}$

Answer 1

${lim}_{x\to 0}\frac{\cos \left(2x^3\right)-\cos \left(5x^3\right)}{x{\sin }^2\left(2x\right){\tan }^3\left(3x\right)}$

${lim}_{x\to 0}\frac{2\sin \left(\right(7/2\left)x^3\right)\sin \left(3x^3/2\right)}{x{\sin }^2\left(2x\right){\tan }^3\left(3x\right)}$

$=\frac{2\times \left(7/2\right)x^3\left(3/2\right)x^3}{x.4x.27x^3}$

$=\frac{7}{18}$