1)[latex] \lim_{x\to 0 } \frac{arctgx}{x} =[/latex] 2)[latex] \lim_{x \to 1/2 } \frac{arcsin(1-2x)}{4 x^{2} -1} =[/latex] 3)[latex] \lim_{x \to 0 } \frac{1-cosx}{ x^{2} } =[/latex] 4)[latex] \lim_{x \to 0 } \frac{tgx-sinx}{ x^{...
1)[latex] \lim_{x\to 0 } \frac{arctgx}{x} =[/latex]
2)[latex] \lim_{x \to 1/2 } \frac{arcsin(1-2x)}{4 x^{2} -1} =[/latex]
3)[latex] \lim_{x \to 0 } \frac{1-cosx}{ x^{2} } =[/latex]
4)[latex] \lim_{x \to 0 } \frac{tgx-sinx}{ x^{3} } =[/latex]
Гость
Ответ(ы) на вопрос:
Гость1)[latex] 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[/latex]
2)[latex] 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}[/latex]
4)[latex] 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} [/latex]
Третий предел равен [latex] \frac{1}{2} [/latex]
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