Question

Вычислите
$displaystyle lim_{xrightarrowpi/2}frac{sin3x+sin x}{cos3x+cos x}$

Answer 1

$lim_{x to frac{ pi }{2} } frac{sin3x+sinx}{cos3x+cosx} = lim_{x to frac{ pi }{2} } frac{2sin frac{3x+x}{2} *cos frac{3x-x}{2} }{2cos frac{3x+x}{2} *cos frac{3x-x}{2}} = \ \ = lim_{x to frac{ pi }{2} } frac{2sin 2x*cos x}{2cos2x *cosx} = lim_{x to frac{ pi }{2} } frac{sin 2x}{cos2x} = lim_{x to frac{ pi }{2} } tg2x=tg(2* frac{ pi }{2} )=\ \tg pi =0$