Задача 53845 Help...
Условие
Решение
[m]=lim_{x\rightarrow \infty}((1+\frac{2}{x+1})^{\frac{x+1}{2}})^{\frac{2}{x+1}\cdot \frac{x}{2}}=e^{lim_{x\rightarrow} \frac{x}{x+1}}=e^{1}=e[/m]
2)
[m]=lim_{x\rightarrow 0}(\frac{sin2x}{2x}\cdot \frac{2x}{3x})=lim_{x\rightarrow 0}\frac{sin2x}{2x}\cdot lim_{x\rightarrow 0}\frac{2x}{3x}=\frac{2}{3}[/m]
3)
[m]=lim_{x\rightarrow 0}\frac{x}{sin^2(\frac{x}{3})}=lim_{x\rightarrow 0}(\frac{x}{sin(\frac{x}{3})}\cdot \frac{1}{sin(\frac{x}{3})})=
=lim_{x\rightarrow 0}\frac{3\cdot \frac{x}{3}}{sin(\frac{x}{3})}\cdot lim_{x\rightarrow 0}\frac{x}{sin(\frac{x}{3})}=3\cdot \frac{1}{0}=\infty[/m]
4)
[m]=lim_{x\rightarrow 0}\frac{tg3x}{x}=lim_{x\rightarrow 0}\frac{tg3x}{3x}\cdot \frac{3}{1}=3[/m]
5)
[m]=lim_{x\rightarrow \infty}((1+\frac{3}{x})^{2x}=lim_{x\rightarrow \infty}((1+\frac{3}{x})^{\frac{x}{3}})^{\frac{3}{x}\cdot 2x}=e^6[/m]
6)
[m]=lim_{x\rightarrow \infty}((1-\frac{3}{x})^{6x}=lim_{x\rightarrow \infty}((1-\frac{3}{x})^{-\frac{x}{3}})^{-\frac{3}{x}\cdot 6x}=e^{-18}[/m]
7)
5)
[m]=lim_{x\rightarrow \infty}((1+\frac{3}{x})^{\frac{x}{2}}=lim_{x\rightarrow \infty}((1+\frac{3}{x})^{\frac{x}{3}})^{\frac{3}{x}\cdot \frac{x}{2}}=e^{\frac{3}{2}}[/m]