Вычислите [latex]\displaystyle \lim_{x\rightarrow\pi/2}\frac{\sin3x+\sin x}{\cos3x+\cos x}[/latex]

[latex] \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[/latex]