1) 2) 3) 4)

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