# lim_(x->0)(5x+tan(3x))/(2x-tan(5x)) = ?

Dec 7, 2016

$- \frac{8}{3}$

#### Explanation:

$\frac{5 x + \sin \frac{3 x}{\cos} \left(3 x\right)}{2 x - \sin \frac{5 x}{\cos} \left(5 x\right)} = \frac{5 x \cos \left(3 x\right) + \sin \left(3 x\right)}{2 x \cos \left(5 x\right) - \sin \left(5 x\right)} \left(\cos \frac{5 x}{\cos} \left(3 x\right)\right) =$

$\left(\frac{5 x}{2 x}\right) \left(\frac{\cos \left(3 x\right) + \sin \frac{3 x}{5 x}}{\cos \left(5 x\right) - \sin \frac{5 x}{2 x}}\right) \left(\cos \frac{5 x}{\cos} \left(3 x\right)\right) =$

$\left(\frac{5}{2}\right) \left(\frac{\cos \left(3 x\right) + \left(\frac{5}{3}\right) \left(\frac{3}{5}\right) \sin \frac{3 x}{5 x}}{\cos \left(5 x\right) - \left(\frac{2}{5}\right) \left(\frac{5}{2}\right) \sin \frac{5 x}{2 x}}\right) \left(\cos \frac{5 x}{\cos} \left(3 x\right)\right) =$

(5/2)((cos(3x)+(3/5)sin(3x)/(3x))/(cos(5x)-(5/2)sin(5x)/(5x))) (cos(5x)/cos(3x))

Now

${\lim}_{x \to 0} \frac{5 x + \tan \left(3 x\right)}{2 x - \tan \left(5 x\right)} = \frac{5}{2} {\lim}_{x \to 0} \left(\frac{\cos \left(3 x\right) + \left(\frac{3}{5}\right) \sin \frac{3 x}{3 x}}{\cos \left(5 x\right) - \left(\frac{5}{2}\right) \sin \frac{5 x}{5 x}}\right) \left({\lim}_{x \to 0} \left(\cos \frac{5 x}{\cos} \left(3 x\right)\right)\right) = \left(\frac{5}{2}\right) \left(\frac{1 + \frac{3}{5}}{1 - \frac{5}{2}}\right) \left(1\right) = - \frac{8}{3}$