Question

Помогите решить лимит
lim(x->0)sin7x/tg2x

Answer 1

$lim_{x to inft0} frac{sin7x}{tg2x} =lim_{x to inft0} frac{sin7x}{ frac{sin2x}{cos2x} } =lim_{x to inft0} frac{cos2x*sin7x}{sin2x} =$

$=lim_{x to inft0} cos2x* lim_{x to inft0} frac{sin7x}{sin2x}=1* lim_{x to inft0} frac{sin7x}{sin2x}=$

$=lim_{x to inft0} frac{ frac{sin7x}{x} }{ frac{sin2x}{x} }= frac{lim_{x to inft0} frac{sin7x}{x} }{lim_{x to inft0} frac{sin2x}{x} } =frac{lim_{x to inft0} frac{7sin7x}{7x} }{lim_{x to inft0} frac{2sin2x}{2x} } =$

$= frac{7 * lim_{x to inft0} frac{sin7x}{7x} }{2 * lim_{x to inft0} frac{sin2x}{2x}} = frac{7*1}{2*1} = frac{7}{2}$

Использовался первый замечательный предел.