Question

Решите пределы! От души

Answer 1

1.
$lim_{n to infty}frac{(3n^2+5)(n-1)!}{(n+1)!}=lim_{n to infty}frac{3n^2+5}{n(n+1)}=lim_{n to infty}frac{3n^2+5}{n^2+n}=3$
2.
$lim_{xto 2}( frac{1}{x-2}- frac{12}{x^3-8})=lim_{xto 2} frac{x^2+2x+4-12}{x^3-8}=lim_{xto 2} frac{x^2+2x-8}{x^3-8}=$$lim_{xto 2} frac{(x-2)(x+4)}{x^3-8}=lim_{xto 2} frac{x+4}{x^2+2x+4}= frac{2+4}{2^2+2*2+4}= frac{1}{2}$
3.
$lim_{x to infty} ( sqrt[3]{8x^3+3x^2+2}-2x)=$$lim_{x to infty} frac{(sqrt[3]{8x^3+3x^2+2}-2x)(sqrt[3]{(8x^3+3x^2+2)^2}+2x*sqrt[3]{8x^3+3x^2+2}+4x^2)}{sqrt[3]{(8x^3+3x^2+2)^2}+2x*sqrt[3]{8x^3+3x^2+2}+4x^2}=$$lim_{x to infty} frac{8x^3+3x^2+2-8x^3}{sqrt[3]{(8x^3+3x^2+2)^2}+2x*sqrt[3]{8x^3+3x^2+2}+4x^2}=$$lim_{x to infty} frac{3x^2+2}{sqrt[3]{(8x^3+3x^2+2)^2}+2x*sqrt[3]{8x^3+3x^2+2}+4x^2}= frac{3}{4+4+4}= frac{1}{4}$
4.
$lim_{x to 0} frac{sin3x-sin7x}{sin5x}=lim_{x to 0}frac{2*sin(-2x)*cos5x}{sin5x}=$$lim_{x to 0}-frac{4x}{5x}= - frac{4}{5}$
5.
$lim_{x to 0} frac{1-e^{6x}}{4x}=lim_{x to 0}-frac{6x}{4x}=- frac{3}{2}$