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4)
![lim_{n to infty} sqrt{n+5}- sqrt{n+1} = \ \
lim_{n to infty} frac{(sqrt{n+5}- sqrt{n+1})*(sqrt{n+5}+ sqrt{n+1} ) }{sqrt{n+5}+ sqrt{n+1} } = \ \
lim_{n to infty} frac{(n+5)- (n+1) }{sqrt{n+5}+ sqrt{n+1} } = \ \
lim_{n to infty} frac{n+5- n-1 }{sqrt{n+5}+ sqrt{n+1} } = \ \
lim_{n to infty} frac{4 }{sqrt{n+5}+ sqrt{n+1} } = frac{4}{infty} =0 lim_{n to infty} sqrt{n+5}- sqrt{n+1} = \ \
lim_{n to infty} frac{(sqrt{n+5}- sqrt{n+1})*(sqrt{n+5}+ sqrt{n+1} ) }{sqrt{n+5}+ sqrt{n+1} } = \ \
lim_{n to infty} frac{(n+5)- (n+1) }{sqrt{n+5}+ sqrt{n+1} } = \ \
lim_{n to infty} frac{n+5- n-1 }{sqrt{n+5}+ sqrt{n+1} } = \ \
lim_{n to infty} frac{4 }{sqrt{n+5}+ sqrt{n+1} } = frac{4}{infty} =0](https://tex.z-dn.net/?f=+lim_%7Bn+to+infty%7D++sqrt%7Bn%2B5%7D-+sqrt%7Bn%2B1%7D+++%3D+%5C++%5C+%0A+lim_%7Bn+to+infty%7D+++frac%7B%28sqrt%7Bn%2B5%7D-+sqrt%7Bn%2B1%7D%29%2A%28sqrt%7Bn%2B5%7D%2B+sqrt%7Bn%2B1%7D+%29+%7D%7Bsqrt%7Bn%2B5%7D%2B+sqrt%7Bn%2B1%7D+%7D+++%3D+%5C++%5C+%0A+lim_%7Bn+to+infty%7D+++frac%7B%28n%2B5%29-+%28n%2B1%29+%7D%7Bsqrt%7Bn%2B5%7D%2B+sqrt%7Bn%2B1%7D+%7D+++%3D+%5C++%5C+%0A+lim_%7Bn+to+infty%7D+++frac%7Bn%2B5-+n-1+%7D%7Bsqrt%7Bn%2B5%7D%2B+sqrt%7Bn%2B1%7D+%7D+++%3D+%5C++%5C+%0A+lim_%7Bn+to+infty%7D+++frac%7B4+%7D%7Bsqrt%7Bn%2B5%7D%2B+sqrt%7Bn%2B1%7D+%7D+++%3D+frac%7B4%7D%7Binfty%7D+%3D0)
5)
![lim_{n to infty} (frac{n-1}{n+2} )^{4n}= \ \
lim_{n to infty} e^{ln((frac{n-1}{n+2} )^{4n})}= \ \
e^{ lim_{n to infty} ln( (frac{n-1}{n+2} )^{4n})}= \ \
e^{ lim_{n to infty} (4n)*ln( frac{n-1}{n+2} )}= \ \
e^{ lim_{n to infty} ( frac{1}{ frac{1}{4n} } )*ln( frac{n-1}{n+2} )}= \ \
e^{ lim_{n to infty} ( frac{ln( frac{n-1}{n+2} )}{ frac{1}{4n} } )}= \ \
e^{ lim_{n to infty} ( frac{(ln( frac{n-1}{n+2} ))'}{ (frac{1}{4n})' } )}= \ \
lim_{n to infty} (frac{n-1}{n+2} )^{4n}= \ \
lim_{n to infty} e^{ln((frac{n-1}{n+2} )^{4n})}= \ \
e^{ lim_{n to infty} ln( (frac{n-1}{n+2} )^{4n})}= \ \
e^{ lim_{n to infty} (4n)*ln( frac{n-1}{n+2} )}= \ \
e^{ lim_{n to infty} ( frac{1}{ frac{1}{4n} } )*ln( frac{n-1}{n+2} )}= \ \
e^{ lim_{n to infty} ( frac{ln( frac{n-1}{n+2} )}{ frac{1}{4n} } )}= \ \
e^{ lim_{n to infty} ( frac{(ln( frac{n-1}{n+2} ))'}{ (frac{1}{4n})' } )}= \ \](https://tex.z-dn.net/?f=+lim_%7Bn+to+infty%7D++%28frac%7Bn-1%7D%7Bn%2B2%7D+%29%5E%7B4n%7D%3D+%5C++%5C+%0A+lim_%7Bn+to+infty%7D++e%5E%7Bln%28%28frac%7Bn-1%7D%7Bn%2B2%7D+%29%5E%7B4n%7D%29%7D%3D+%5C++%5C+%0Ae%5E%7B+lim_%7Bn+to+infty%7D+ln%28+%28frac%7Bn-1%7D%7Bn%2B2%7D+%29%5E%7B4n%7D%29%7D%3D++%5C++%5C+%0Ae%5E%7B+lim_%7Bn+to+infty%7D+%284n%29%2Aln%28+frac%7Bn-1%7D%7Bn%2B2%7D+%29%7D%3D++%5C++%5C+%0Ae%5E%7B+lim_%7Bn+to+infty%7D+%28+frac%7B1%7D%7B+frac%7B1%7D%7B4n%7D+%7D+%29%2Aln%28+frac%7Bn-1%7D%7Bn%2B2%7D+%29%7D%3D++%5C++%5C+%0Ae%5E%7B+lim_%7Bn+to+infty%7D+%28+frac%7Bln%28+frac%7Bn-1%7D%7Bn%2B2%7D+%29%7D%7B+frac%7B1%7D%7B4n%7D+%7D+%29%7D%3D++%5C++%5C+%0Ae%5E%7B+lim_%7Bn+to+infty%7D+%28+frac%7B%28ln%28+frac%7Bn-1%7D%7Bn%2B2%7D+%29%29%27%7D%7B+%28frac%7B1%7D%7B4n%7D%29%27+%7D+%29%7D%3D++%5C++%5C+%0A%0A)
![e^{ lim_{n to infty} frac{( frac{n+2}{n-1})( frac{n-1}{n+2} )'}{ -frac{(4n)'}{(4n)^{2}} } }= \ \
e^{ lim_{n to infty} frac{( frac{n+2}{n-1})( frac{(n+2)-(n-1)}{(n+2)^{2}} }{ -frac{4}{4*4n^{2}} } }= \ \
e^{ lim_{n to infty} frac{ frac{3}{(n-1)(n+2)} }{ -frac{1}{4n^{2}} } }= \ \
e^{ lim_{n to infty} - frac{3*4 n^{2}}{n^{2}+n-2}} = \ \
e^{-12*{ lim_{n to infty} frac{n^{2}}{n^{2}+n-2} }}= \ \
e^{ lim_{n to infty} frac{( frac{n+2}{n-1})( frac{n-1}{n+2} )'}{ -frac{(4n)'}{(4n)^{2}} } }= \ \
e^{ lim_{n to infty} frac{( frac{n+2}{n-1})( frac{(n+2)-(n-1)}{(n+2)^{2}} }{ -frac{4}{4*4n^{2}} } }= \ \
e^{ lim_{n to infty} frac{ frac{3}{(n-1)(n+2)} }{ -frac{1}{4n^{2}} } }= \ \
e^{ lim_{n to infty} - frac{3*4 n^{2}}{n^{2}+n-2}} = \ \
e^{-12*{ lim_{n to infty} frac{n^{2}}{n^{2}+n-2} }}= \ \](https://tex.z-dn.net/?f=e%5E%7B+lim_%7Bn+to+infty%7D++frac%7B%28+frac%7Bn%2B2%7D%7Bn-1%7D%29%28+frac%7Bn-1%7D%7Bn%2B2%7D+%29%27%7D%7B+-frac%7B%284n%29%27%7D%7B%284n%29%5E%7B2%7D%7D+%7D+%7D%3D++%5C++%5C+%0Ae%5E%7B+lim_%7Bn+to+infty%7D++frac%7B%28+frac%7Bn%2B2%7D%7Bn-1%7D%29%28+frac%7B%28n%2B2%29-%28n-1%29%7D%7B%28n%2B2%29%5E%7B2%7D%7D+%7D%7B+-frac%7B4%7D%7B4%2A4n%5E%7B2%7D%7D+%7D+%7D%3D++%5C++%5C+%0Ae%5E%7B+lim_%7Bn+to+infty%7D++frac%7B+frac%7B3%7D%7B%28n-1%29%28n%2B2%29%7D+%7D%7B+-frac%7B1%7D%7B4n%5E%7B2%7D%7D+%7D+%7D%3D++%5C++%5C+%0Ae%5E%7B+lim_%7Bn+to+infty%7D+-++frac%7B3%2A4+n%5E%7B2%7D%7D%7Bn%5E%7B2%7D%2Bn-2%7D%7D+%3D+%5C++%5C+%0Ae%5E%7B-12%2A%7B+lim_%7Bn+to+infty%7D++frac%7Bn%5E%7B2%7D%7D%7Bn%5E%7B2%7D%2Bn-2%7D+%7D%7D%3D+%5C++%5C+%0A)
5)
Ответ дал:
0
Спасибо. 7, 8 можете помочь сделать?
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