1)[latex] lim_{xo 0 } frac{arctgx}{x} =[/latex] 2)[latex] lim_{x o 1/2 } frac{arcsin(1-2x)}{4 x^{2} -1} =[/latex] 3)[latex] lim_{x o 0 } frac{1-cosx}{ x^{2} } =[/latex] 4)[latex] lim_{x o 0 } frac{tgx-sinx}{ x^{3} } =[/latex]

1)[latex] lim _{x- extgreater 0} frac{arctgx}{x} = \ arctgx=y\ tgy=x\ lim _{y- extgreater 0} frac{arctgx}{x} = frac{y}{tgy} = frac{y*cosy}{siny} = 1*1=1[/latex]2)[latex] lim _{x- extgreater 0.5} frac{arcsin(1-2x)}{4x^2-1} = lim _{x- extgreater 0.5} frac{arcsin(1-2x)}{-(1-2x)(2x+1)} \ 1-2x=y\ lim_{y- extgreater 0} frac{arcsiny}{-2y} = - 1*frac{1}{2}=-frac{1}{2}[/latex]   4)[latex] lim _ {x- extgreater 0 } frac{tgx-sinx}{x^3} = \ lim_{x- extgreater 0} frac{ ( 2sinx-sin2x )}{(2x^3*cosx) } = frac{8*sin^3(frac{x}{ } )*cos (frac{x}{2})}{2x^3*cosx} = \ lim _{x- extgreater 0} frac{8*sin^3frac{x}{3}*1}{1*8*(frac{x}{2})^3*2} = frac{1}{2} [/latex] Третий предел равен [latex] frac{1}{2} [/latex]