Решение на рисунке !!!!!!!!!!!
4 м>4 дм 60 см>5 дм
100 дм2=1м2 30дм=3м
1) 72-50=22 (ч)- стоят
Обратная задача
2) 72-22=50 (ч)- сидят
4)
![\lim_{x \to \infty} (\frac{7x+3}{7x-1})^{2x} = \lim_{x \to \infty} (1+\frac{4}{7x-1})^{2x}](https://tex.z-dn.net/?f=+%5Clim_%7Bx+%5Cto+%5Cinfty%7D+%28%5Cfrac%7B7x%2B3%7D%7B7x-1%7D%29%5E%7B2x%7D+%3D++%5Clim_%7Bx+%5Cto+%5Cinfty%7D+%281%2B%5Cfrac%7B4%7D%7B7x-1%7D%29%5E%7B2x%7D)
Так как
![\lim_{x \to \infty} (\frac{4}{7x-1}) =0](https://tex.z-dn.net/?f=%5Clim_%7Bx+%5Cto+%5Cinfty%7D+%28%5Cfrac%7B4%7D%7B7x-1%7D%29+%3D0)
То
![\lim_{x \to \infty} (1+\frac{1}{\frac{7x-1}{4}})^{\frac{7x-1}{4}} = \lim_{x \to \infty} (1+\frac{4}{7x-1})^{\frac{7x-1}{4}} = e](https://tex.z-dn.net/?f=+%5Clim_%7Bx+%5Cto+%5Cinfty%7D+%281%2B%5Cfrac%7B1%7D%7B%5Cfrac%7B7x-1%7D%7B4%7D%7D%29%5E%7B%5Cfrac%7B7x-1%7D%7B4%7D%7D+%3D++%5Clim_%7Bx+%5Cto+%5Cinfty%7D+%281%2B%5Cfrac%7B4%7D%7B7x-1%7D%29%5E%7B%5Cfrac%7B7x-1%7D%7B4%7D%7D+%3D+e)
Тогда:
![\lim_{x \to \infty} (1+\frac{1}{\frac{7x-1}{4}})^{2x} = \lim_{x \to \infty} ((1+\frac{1}{\frac{7x-1}{4}})^{\frac{7x-1}{4}})^{\frac{2x * 4}{7x-1}} = \\ = \lim_{x \to \infty} ((1+\frac{1}{\frac{7x-1}{4}})^{\frac{7x-1}{4}})^{\frac{\frac{2x}{x} * 4}{\frac{7x}{x}-\frac{1}{x}}} = e^{\frac{8}{7}}](https://tex.z-dn.net/?f=+%5Clim_%7Bx+%5Cto+%5Cinfty%7D+%281%2B%5Cfrac%7B1%7D%7B%5Cfrac%7B7x-1%7D%7B4%7D%7D%29%5E%7B2x%7D+%3D++%5Clim_%7Bx+%5Cto+%5Cinfty%7D+%28%281%2B%5Cfrac%7B1%7D%7B%5Cfrac%7B7x-1%7D%7B4%7D%7D%29%5E%7B%5Cfrac%7B7x-1%7D%7B4%7D%7D%29%5E%7B%5Cfrac%7B2x+%2A+4%7D%7B7x-1%7D%7D+%3D+%5C%5C+%3D++%5Clim_%7Bx+%5Cto+%5Cinfty%7D+%28%281%2B%5Cfrac%7B1%7D%7B%5Cfrac%7B7x-1%7D%7B4%7D%7D%29%5E%7B%5Cfrac%7B7x-1%7D%7B4%7D%7D%29%5E%7B%5Cfrac%7B%5Cfrac%7B2x%7D%7Bx%7D+%2A+4%7D%7B%5Cfrac%7B7x%7D%7Bx%7D-%5Cfrac%7B1%7D%7Bx%7D%7D%7D+%3D+e%5E%7B%5Cfrac%7B8%7D%7B7%7D%7D)
5)
![\lim_{x \to 0} \frac{sin^3x}{3x^3} = \lim_{x \to 0} (\frac{1}{3} * (\frac{sin x}{x})^3)](https://tex.z-dn.net/?f=+%5Clim_%7Bx+%5Cto+0%7D+%5Cfrac%7Bsin%5E3x%7D%7B3x%5E3%7D+%3D+%5Clim_%7Bx+%5Cto+0%7D+%28%5Cfrac%7B1%7D%7B3%7D+%2A+%28%5Cfrac%7Bsin+x%7D%7Bx%7D%29%5E3%29)
По первому замечательному пределу:
![\lim_{x \to 0} \frac{sin x}{x} = 1](https://tex.z-dn.net/?f=+%5Clim_%7Bx+%5Cto+0%7D+%5Cfrac%7Bsin+x%7D%7Bx%7D+%3D+1+)
Тогда получаем:
![\lim_{x \to 0} (\frac{1}{3} * (\frac{sin x}{x})^3) = \frac {1}{3} * 1 = \frac {1}{3}](https://tex.z-dn.net/?f=%5Clim_%7Bx+%5Cto+0%7D+%28%5Cfrac%7B1%7D%7B3%7D+%2A+%28%5Cfrac%7Bsin+x%7D%7Bx%7D%29%5E3%29+%3D+%5Cfrac+%7B1%7D%7B3%7D+%2A+1+%3D+%5Cfrac+%7B1%7D%7B3%7D)
6)
![\lim_{x \to \frac{\pi}{2} } \frac{ctg x}{\pi - 2x} = \lim_{x \to \frac{\pi}{2}}\frac{cos x}{sin x(\pi - 2x)}](https://tex.z-dn.net/?f=+%5Clim_%7Bx+%5Cto++%5Cfrac%7B%5Cpi%7D%7B2%7D+%7D+%5Cfrac%7Bctg+x%7D%7B%5Cpi+-+2x%7D+%3D++%5Clim_%7Bx+%5Cto++%5Cfrac%7B%5Cpi%7D%7B2%7D%7D%5Cfrac%7Bcos+x%7D%7Bsin+x%28%5Cpi+-+2x%29%7D)
При подстановке значения x в выражение получаем неопределенность
![\frac{0}{0}](https://tex.z-dn.net/?f=%5Cfrac%7B0%7D%7B0%7D)
.
Сделаем замену:
![t = \pi - 2x \\ t \to 0](https://tex.z-dn.net/?f=t+%3D+%5Cpi+-+2x+%5C%5C+t+%5Cto+0)
Тогда:
![x = \frac{\pi - t}{2}](https://tex.z-dn.net/?f=x+%3D+%5Cfrac%7B%5Cpi+-+t%7D%7B2%7D)
Выполним преобразования:
![\lim_{t \to 0} \frac{cos(\frac{\pi - t}{2})}{sin(\frac{\pi-t}{2})t} = \lim_{t \to 0} \frac{cos\frac{\pi}{2}cos\frac{t}{2} + sin\frac{\pi}{2}sin\frac{t}{2}}{t(sin\frac{\pi}{2}sin\frac{t}{2} - cos\frac{\pi}{2}sin\frac{t}{2})} = \\ \lim_{t \to 0} \frac{cos\frac{\pi}{2}cos\frac{t}{2}}{t(sin\frac{\pi}{2}sin\frac{t}{2} - cos\frac{\pi}{2}sin\frac{t}{2})} + \lim_{t \to 0} \frac{sib\frac{\pi}{2}sin\frac{t}{2}*\frac{1}{2}}{t(sin\frac{\pi}{2}sin\frac{t}{2} - cos\frac{\pi}{2}sin\frac{t}{2})*\frac{1}{2}}](https://tex.z-dn.net/?f=%5Clim_%7Bt+%5Cto+0%7D+%5Cfrac%7Bcos%28%5Cfrac%7B%5Cpi+-+t%7D%7B2%7D%29%7D%7Bsin%28%5Cfrac%7B%5Cpi-t%7D%7B2%7D%29t%7D+%3D+%5Clim_%7Bt+%5Cto+0%7D+%5Cfrac%7Bcos%5Cfrac%7B%5Cpi%7D%7B2%7Dcos%5Cfrac%7Bt%7D%7B2%7D+%2B+sin%5Cfrac%7B%5Cpi%7D%7B2%7Dsin%5Cfrac%7Bt%7D%7B2%7D%7D%7Bt%28sin%5Cfrac%7B%5Cpi%7D%7B2%7Dsin%5Cfrac%7Bt%7D%7B2%7D+-+cos%5Cfrac%7B%5Cpi%7D%7B2%7Dsin%5Cfrac%7Bt%7D%7B2%7D%29%7D+%3D+%5C%5C++%5Clim_%7Bt+%5Cto+0%7D+%5Cfrac%7Bcos%5Cfrac%7B%5Cpi%7D%7B2%7Dcos%5Cfrac%7Bt%7D%7B2%7D%7D%7Bt%28sin%5Cfrac%7B%5Cpi%7D%7B2%7Dsin%5Cfrac%7Bt%7D%7B2%7D+-+cos%5Cfrac%7B%5Cpi%7D%7B2%7Dsin%5Cfrac%7Bt%7D%7B2%7D%29%7D+%2B+%5Clim_%7Bt+%5Cto+0%7D+%5Cfrac%7Bsib%5Cfrac%7B%5Cpi%7D%7B2%7Dsin%5Cfrac%7Bt%7D%7B2%7D%2A%5Cfrac%7B1%7D%7B2%7D%7D%7Bt%28sin%5Cfrac%7B%5Cpi%7D%7B2%7Dsin%5Cfrac%7Bt%7D%7B2%7D+-+cos%5Cfrac%7B%5Cpi%7D%7B2%7Dsin%5Cfrac%7Bt%7D%7B2%7D%29%2A%5Cfrac%7B1%7D%7B2%7D%7D+)
![= 0 + \frac{1 * \frac{1}{2}}{(1 - 0)*1} = \frac{1}{2}](https://tex.z-dn.net/?f=%3D+0+%2B++%5Cfrac%7B1+%2A+%5Cfrac%7B1%7D%7B2%7D%7D%7B%281+-+0%29%2A1%7D+%3D+%5Cfrac%7B1%7D%7B2%7D)
<span>75у-156= 23у
75у-23у=156
52у=156
у=156:52
у=3
</span>