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В знаменателе разница кубов, раскладываем её.
![lim_{n to infty} frac{(3-4n)^2}{(n-3)^3 - (n+3)^3} = lim_{n to infty}frac{(3-4n)^2}{(n-3 - n -3)((n+3)^2 + (n+3)(n-3) + (n-3)^2} lim_{n to infty} frac{(3-4n)^2}{(n-3)^3 - (n+3)^3} = lim_{n to infty}frac{(3-4n)^2}{(n-3 - n -3)((n+3)^2 + (n+3)(n-3) + (n-3)^2}](https://tex.z-dn.net/?f=+lim_%7Bn+to+infty%7D++frac%7B%283-4n%29%5E2%7D%7B%28n-3%29%5E3+-+%28n%2B3%29%5E3%7D+%3D+lim_%7Bn+to+infty%7Dfrac%7B%283-4n%29%5E2%7D%7B%28n-3+-+n+-3%29%28%28n%2B3%29%5E2+%2B+%28n%2B3%29%28n-3%29+%2B+%28n-3%29%5E2%7D+)
![lim_{n to infty}frac{(3-4n)^2}{(n-3 - n -3)((n+3)^2 + (n+3)(n-3) + (n-3)^2} = lim_{n to infty} frac{(3-4n)^2}{-6(3n^2 + 9)} lim_{n to infty}frac{(3-4n)^2}{(n-3 - n -3)((n+3)^2 + (n+3)(n-3) + (n-3)^2} = lim_{n to infty} frac{(3-4n)^2}{-6(3n^2 + 9)}](https://tex.z-dn.net/?f=lim_%7Bn+to+infty%7Dfrac%7B%283-4n%29%5E2%7D%7B%28n-3+-+n+-3%29%28%28n%2B3%29%5E2+%2B+%28n%2B3%29%28n-3%29+%2B+%28n-3%29%5E2%7D++%3D+lim_%7Bn+to+infty%7D++frac%7B%283-4n%29%5E2%7D%7B-6%283n%5E2+%2B+9%29%7D++)
![lim_{n to infty} frac{(3-4n)^2}{-6(3n^2 + 9)} = lim_{n to infty} frac{(3-4n)^2}{-18n^2 - 54} = lim_{n to infty} frac{9 - 24n + 16n^2}{-18n^2 - 54} lim_{n to infty} frac{(3-4n)^2}{-6(3n^2 + 9)} = lim_{n to infty} frac{(3-4n)^2}{-18n^2 - 54} = lim_{n to infty} frac{9 - 24n + 16n^2}{-18n^2 - 54}](https://tex.z-dn.net/?f=lim_%7Bn+to+infty%7D++frac%7B%283-4n%29%5E2%7D%7B-6%283n%5E2+%2B+9%29%7D++%3D+lim_%7Bn+to+infty%7D+frac%7B%283-4n%29%5E2%7D%7B-18n%5E2+-+54%7D+%3D+lim_%7Bn+to+infty%7D++frac%7B9+-+24n+%2B+16n%5E2%7D%7B-18n%5E2+-+54%7D++)
![lim_{n to infty} frac{9 - 24n + 16n^2}{-18n^2 - 54} =lim_{n to infty} frac{n^2(16 - 24/n + 9/n^2)}{n^2(-18 - 54/n^2)} = -frac{16}{18} = - frac{8}{9} lim_{n to infty} frac{9 - 24n + 16n^2}{-18n^2 - 54} =lim_{n to infty} frac{n^2(16 - 24/n + 9/n^2)}{n^2(-18 - 54/n^2)} = -frac{16}{18} = - frac{8}{9}](https://tex.z-dn.net/?f=lim_%7Bn+to+infty%7D++frac%7B9+-+24n+%2B+16n%5E2%7D%7B-18n%5E2+-+54%7D++%3Dlim_%7Bn+to+infty%7D++frac%7Bn%5E2%2816+-+24%2Fn+%2B+9%2Fn%5E2%29%7D%7Bn%5E2%28-18+-+54%2Fn%5E2%29%7D+%3D++-frac%7B16%7D%7B18%7D+%3D+-+frac%7B8%7D%7B9%7D++)
Домножаем на сопряженное, чтобы получилась разность квадратов в числителе.
![lim_{n to infty} sqrt{n^2 + 2n} - sqrt {n^2 - 2n +3} lim_{n to infty} sqrt{n^2 + 2n} - sqrt {n^2 - 2n +3}](https://tex.z-dn.net/?f=+lim_%7Bn+to+infty%7D++sqrt%7Bn%5E2+%2B+2n%7D+-+++sqrt+%7Bn%5E2+-+2n+%2B3%7D++)
![lim_{n to infty} sqrt{n^2 + 2n} -sqrt {n^2 - 2n +3} frac{sqrt{n^2 + 2n} +sqrt {n^2 - 2n +3}}{sqrt{n^2 + 2n} +sqrt {n^2 - 2n +3}} lim_{n to infty} sqrt{n^2 + 2n} -sqrt {n^2 - 2n +3} frac{sqrt{n^2 + 2n} +sqrt {n^2 - 2n +3}}{sqrt{n^2 + 2n} +sqrt {n^2 - 2n +3}}](https://tex.z-dn.net/?f=lim_%7Bn+to+infty%7D+sqrt%7Bn%5E2+%2B+2n%7D+-sqrt+%7Bn%5E2+-+2n+%2B3%7D+frac%7Bsqrt%7Bn%5E2+%2B+2n%7D+%2Bsqrt+%7Bn%5E2+-+2n+%2B3%7D%7D%7Bsqrt%7Bn%5E2+%2B+2n%7D+%2Bsqrt+%7Bn%5E2+-+2n+%2B3%7D%7D++)
![lim_{n to infty} frac{n^2 + 2n - n^2 -2n + 3}{sqrt{n^2 + 2n} +sqrt {n^2 - 2n +3}} = lim_{n to infty} frac{3}{sqrt{n^2 + 2n} +sqrt {n^2 - 2n +3}} lim_{n to infty} frac{n^2 + 2n - n^2 -2n + 3}{sqrt{n^2 + 2n} +sqrt {n^2 - 2n +3}} = lim_{n to infty} frac{3}{sqrt{n^2 + 2n} +sqrt {n^2 - 2n +3}}](https://tex.z-dn.net/?f=lim_%7Bn+to+infty%7D+frac%7Bn%5E2+%2B+2n+-+n%5E2+-2n+%2B+3%7D%7Bsqrt%7Bn%5E2+%2B+2n%7D+%2Bsqrt+%7Bn%5E2+-+2n+%2B3%7D%7D+%3D+lim_%7Bn+to+infty%7D++frac%7B3%7D%7Bsqrt%7Bn%5E2+%2B+2n%7D+%2Bsqrt+%7Bn%5E2+-+2n+%2B3%7D%7D+)
![lim_{n to infty} frac{3}{sqrt{n^2 + 2n} +sqrt {n^2 - 2n +3}} = lim_{n to infty} frac{3}{infty} = 0 lim_{n to infty} frac{3}{sqrt{n^2 + 2n} +sqrt {n^2 - 2n +3}} = lim_{n to infty} frac{3}{infty} = 0](https://tex.z-dn.net/?f=lim_%7Bn+to+infty%7D++frac%7B3%7D%7Bsqrt%7Bn%5E2+%2B+2n%7D+%2Bsqrt+%7Bn%5E2+-+2n+%2B3%7D%7D++%3D+lim_%7Bn+to+infty%7D++frac%7B3%7D%7Binfty%7D+%3D+0)
Домножаем на сопряженное, чтобы получилась разность квадратов в числителе.
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