6<3b-a<8 vot pravel'no bratiw
![1) \lim_{x \to \infty} ( \sqrt{3x+1} - \sqrt{x+5} )= \{\infty-\infty \}= \\ \\ \lim_{x \to \infty} \frac{( \sqrt{3x+1} - \sqrt{x+5})( \sqrt{3x+1} - \sqrt{x+5})}{( \sqrt{3x+1} - \sqrt{x+5})} = \lim_{x \to \infty} \frac{3x+1-x-5}{ \sqrt{3x+1} +\sqrt{x+5}} = \\ \\ =\lim_{x \to \infty} \frac{2x-4}{ \sqrt{3x+1} +\sqrt{x+5}} =\{ \frac{\infty}{\infty} \}= \lim_{x \to \infty} \frac{ \frac{2x}{x} - \frac{4}{x} }{ \sqrt{ \frac{3x+1}{x^2} }+ \sqrt{ \frac{x+5}{x^2} } } =](https://tex.z-dn.net/?f=1%29+%5Clim_%7Bx+%5Cto+%5Cinfty%7D+%28+%5Csqrt%7B3x%2B1%7D+-+%5Csqrt%7Bx%2B5%7D+%29%3D+%5C%7B%5Cinfty-%5Cinfty+%5C%7D%3D++%5C%5C++%5C%5C+%5Clim_%7Bx+%5Cto+%5Cinfty%7D+%5Cfrac%7B%28+%5Csqrt%7B3x%2B1%7D+-+%5Csqrt%7Bx%2B5%7D%29%28+%5Csqrt%7B3x%2B1%7D+-+%5Csqrt%7Bx%2B5%7D%29%7D%7B%28+%5Csqrt%7B3x%2B1%7D+-+%5Csqrt%7Bx%2B5%7D%29%7D+%3D+%5Clim_%7Bx+%5Cto+%5Cinfty%7D++%5Cfrac%7B3x%2B1-x-5%7D%7B+%5Csqrt%7B3x%2B1%7D+%2B%5Csqrt%7Bx%2B5%7D%7D+%3D+%5C%5C+%5C%5C+%3D%5Clim_%7Bx+%5Cto+%5Cinfty%7D++%5Cfrac%7B2x-4%7D%7B+%5Csqrt%7B3x%2B1%7D+%2B%5Csqrt%7Bx%2B5%7D%7D+%3D%5C%7B+%5Cfrac%7B%5Cinfty%7D%7B%5Cinfty%7D++%5C%7D%3D+%5Clim_%7Bx+%5Cto+%5Cinfty%7D++%5Cfrac%7B+%5Cfrac%7B2x%7D%7Bx%7D+-++%5Cfrac%7B4%7D%7Bx%7D+%7D%7B+%5Csqrt%7B+%5Cfrac%7B3x%2B1%7D%7Bx%5E2%7D+%7D%2B+%5Csqrt%7B+%5Cfrac%7Bx%2B5%7D%7Bx%5E2%7D+%7D++%7D+%3D)
![= \lim_{x \to +\infty} \frac{2- \frac{4}{x} }{ \sqrt{ \frac{3}{x}+ \frac{1}{x^2} }+ \sqrt{ \frac{1}{x}+ \frac{5}{x^2} } } = \frac{2- \frac{4}{\infty} }{ \sqrt{ \frac{3}{\infty}+ \frac{1}{\infty^2}} + \sqrt{ \frac{1}{\infty}+ \frac{5}{\infty^2} } } =\\ \\ = \frac{2-0}{ \sqrt{0+0}+ \sqrt{0+0} } = \frac{2}{0} =\infty](https://tex.z-dn.net/?f=%3D++%5Clim_%7Bx+%5Cto+%2B%5Cinfty%7D++%5Cfrac%7B2-++%5Cfrac%7B4%7D%7Bx%7D+%7D%7B+%5Csqrt%7B+%5Cfrac%7B3%7D%7Bx%7D%2B+%5Cfrac%7B1%7D%7Bx%5E2%7D+%7D%2B+%5Csqrt%7B+%5Cfrac%7B1%7D%7Bx%7D%2B+%5Cfrac%7B5%7D%7Bx%5E2%7D++%7D++%7D+%3D++%5Cfrac%7B2-++%5Cfrac%7B4%7D%7B%5Cinfty%7D+%7D%7B+%5Csqrt%7B+%5Cfrac%7B3%7D%7B%5Cinfty%7D%2B+%5Cfrac%7B1%7D%7B%5Cinfty%5E2%7D%7D+%2B+%5Csqrt%7B+%5Cfrac%7B1%7D%7B%5Cinfty%7D%2B+%5Cfrac%7B5%7D%7B%5Cinfty%5E2%7D++%7D++%7D+%3D%5C%5C+%5C%5C+%3D+%5Cfrac%7B2-0%7D%7B+%5Csqrt%7B0%2B0%7D%2B++%5Csqrt%7B0%2B0%7D+++%7D+%3D+%5Cfrac%7B2%7D%7B0%7D+%3D%5Cinfty)
![2) \ \lim_{x \to 0}( \frac{2x+1}{x+1} )^{ \frac{1}{x}} =\{1^\infty \}=\lim_{x \to 0}( \frac{x+x+1}{x+1} )^{ \frac{1}{x}}=\lim_{x \to 0}( 1+ \frac{x}{x+1} )^{ \frac{1}{x}}= \\ \\ =\lim_{x \to 0}( 1+ \frac{x}{x+1} )^{ \frac{x+1}{x}* \frac{x}{x+1} * \frac{1}{x}}= \lim_{x \to 0} (e^{\frac{x}{x+1} * \frac{1}{x}}) =\lim_{x \to 0} (e^{\frac{1}{x+1} }) = \\ \\ = e^{ \lim_{x \to0} ( \frac{1}{x+1})}=e^{ \frac{1}{0+1} } =e^1=e](https://tex.z-dn.net/?f=2%29+%5C++%5Clim_%7Bx+%5Cto+0%7D%28+%5Cfrac%7B2x%2B1%7D%7Bx%2B1%7D+%29%5E%7B+%5Cfrac%7B1%7D%7Bx%7D%7D+%3D%5C%7B1%5E%5Cinfty+%5C%7D%3D%5Clim_%7Bx+%5Cto+0%7D%28+%5Cfrac%7Bx%2Bx%2B1%7D%7Bx%2B1%7D+%29%5E%7B+%5Cfrac%7B1%7D%7Bx%7D%7D%3D%5Clim_%7Bx+%5Cto+0%7D%28++1%2B+%5Cfrac%7Bx%7D%7Bx%2B1%7D+%29%5E%7B+%5Cfrac%7B1%7D%7Bx%7D%7D%3D+%5C%5C+%5C%5C+%3D%5Clim_%7Bx+%5Cto+0%7D%28++1%2B+%5Cfrac%7Bx%7D%7Bx%2B1%7D+%29%5E%7B++%5Cfrac%7Bx%2B1%7D%7Bx%7D%2A+%5Cfrac%7Bx%7D%7Bx%2B1%7D+%2A+%5Cfrac%7B1%7D%7Bx%7D%7D%3D+%5Clim_%7Bx+%5Cto+0%7D+%28e%5E%7B%5Cfrac%7Bx%7D%7Bx%2B1%7D+%2A+%5Cfrac%7B1%7D%7Bx%7D%7D%29+%3D%5Clim_%7Bx+%5Cto+0%7D+%28e%5E%7B%5Cfrac%7B1%7D%7Bx%2B1%7D+%7D%29+%3D+%5C%5C+%5C%5C+%3D+e%5E%7B+%5Clim_%7Bx+%5Cto0%7D+%28+%5Cfrac%7B1%7D%7Bx%2B1%7D%29%7D%3De%5E%7B+%5Cfrac%7B1%7D%7B0%2B1%7D+%7D+%3De%5E1%3De)
![4) \lim_{x \to \frac{ \pi }{2} } (sinx)^{tgx}=\{1^\infty \}= \lim_{x \to \frac{ \pi }{2} }{(1+sinx-1)^{ \frac{1}{sinx -1} *(sinx-1)*tgx} }\\ \\ = \lim_{x \to \frac{ \pi }{2} }(e^{(sinx-1)tgx})=e^{ \lim_{x \to \frac{ \pi }{2} }(sinx-1)tgx](https://tex.z-dn.net/?f=4%29++%5Clim_%7Bx+%5Cto++%5Cfrac%7B+%5Cpi+%7D%7B2%7D+%7D+%28sinx%29%5E%7Btgx%7D%3D%5C%7B1%5E%5Cinfty+%5C%7D%3D+%5Clim_%7Bx+%5Cto++%5Cfrac%7B+%5Cpi+%7D%7B2%7D+%7D%7B%281%2Bsinx-1%29%5E%7B+%5Cfrac%7B1%7D%7Bsinx++-1%7D+%2A%28sinx-1%29%2Atgx%7D+%7D%5C%5C++%5C%5C+%3D+%5Clim_%7Bx+%5Cto++%5Cfrac%7B+%5Cpi+%7D%7B2%7D+%7D%28e%5E%7B%28sinx-1%29tgx%7D%29%3De%5E%7B+%5Clim_%7Bx+%5Cto++%5Cfrac%7B+%5Cpi+%7D%7B2%7D+%7D%28sinx-1%29tgx)
Найдем отдельно предел показателя, произведя замену х→π/2, на
х-π/2 →0:
![\lim_{x \to \frac{ \pi }{2} }(sinx-1)tgx= \left[\begin{array}{c}x- \frac{ \pi }{2} =t \\x= \frac{ \pi }{2}+t&t \to 0 \\ \end{array}\right] = \\ \\ \lim_{t \to0} (sin(\frac{ \pi }{2}+t)-1)tg(\frac{ \pi }{2}+t)= \lim_{t \to 0} (cost-1)*(-ctgt) = \\ \\ =\lim_{t \to 0}- (1-cost)*(-ctgt) =\lim_{t \to 0} (1-cost)* \frac{1}{tgt} =](https://tex.z-dn.net/?f=%5Clim_%7Bx+%5Cto+%5Cfrac%7B+%5Cpi+%7D%7B2%7D+%7D%28sinx-1%29tgx%3D+++%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx-+%5Cfrac%7B+%5Cpi+%7D%7B2%7D+%3Dt+%5C%5Cx%3D+%5Cfrac%7B+%5Cpi+%7D%7B2%7D%2Bt%26t+%5Cto+0+%5C%5C++%5Cend%7Barray%7D%5Cright%5D+%3D+%5C%5C++%5C%5C+++%5Clim_%7Bt+%5Cto0%7D+%28sin%28%5Cfrac%7B+%5Cpi+%7D%7B2%7D%2Bt%29-1%29tg%28%5Cfrac%7B+%5Cpi+%7D%7B2%7D%2Bt%29%3D+%5Clim_%7Bt+%5Cto+0%7D+%28cost-1%29%2A%28-ctgt%29+%3D++%5C%5C++%5C%5C++%3D%5Clim_%7Bt+%5Cto+0%7D-+%281-cost%29%2A%28-ctgt%29+%3D%5Clim_%7Bt+%5Cto+0%7D+%281-cost%29%2A+%5Cfrac%7B1%7D%7Btgt%7D+%3D)
Далее пользуемся таблицей эквивалентности: заменяем
1-cost на t²/2
tgt на t
![= \lim_{t \to 0} \frac{t^2}{2} * \frac{1}{t} =\lim_{t \to 0} \frac{t}{2} = \frac{0}{2} =0](https://tex.z-dn.net/?f=%3D+%5Clim_%7Bt+%5Cto+0%7D++%5Cfrac%7Bt%5E2%7D%7B2%7D+%2A+%5Cfrac%7B1%7D%7Bt%7D+%3D%5Clim_%7Bt+%5Cto+0%7D++%5Cfrac%7Bt%7D%7B2%7D++%3D+%5Cfrac%7B0%7D%7B2%7D+%3D0)
![e^ \lim_{x \to \frac{ \pi }{2} } (sinx-1)tgx}=e^0=1](https://tex.z-dn.net/?f=e%5E+%5Clim_%7Bx+%5Cto++%5Cfrac%7B+%5Cpi+%7D%7B2%7D+%7D+%28sinx-1%29tgx%7D%3De%5E0%3D1)
Task/24790597
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Решить уравнение 4sin²x-15cos²<span>x=4 sinx cosx
</span>4sin²x- 4 sinx cosx -15cos²x = 0 || : cos²x ≠0
4tq²x - 4tgx -15 =0 ;
квадратное уравнение относительно tgx
D/4 =2² -4*(-15) =4+60 =64 =8²
a) tqx₁ =(2 -8)/4 = -3/2 ⇒ x₁ = - arctq1,5 + π*n ,n ∈Z
b) tqx₂ =(2+8)/4 =5/2 ⇒ x<span>₂</span> = arctq2,5 + π*n ,n ∈Z