Question

Вычислите предел функции

Answer 1

Ответ:

$6.lim_{n to infty} frac{n^3+6n-5}{10n^3-8n^2+2}=0,1$

$7.lim_{x to 4} frac{x^2-16}{x^2-5x+4}=2frac{2}{3}$

$8. lim_{x to 2} frac{sqrt{x-1}-1 }{sqrt{x}-sqrt{2} }= sqrt{2}$

Объяснение:

6. вычислите предел последовательности $lim_{n to infty} frac{n^3+6n-5}{10n^3-8n^2+2}$

$lim_{n to infty} frac{n^3+6n-5}{10n^3-8n^2+2}= lim_{n to infty} frac{n^3(1+frac{6}{n^2} -frac{5}{n^3})}{n^3(10-frac{8}{n} +frac{2}{n^3})}=lim_{n to infty} frac{1+frac{6}{n^2} -frac{5}{n^3}}{10-frac{8}{n} +frac{2}{n^3}}=frac{1+0-0}{10-0+0}=frac{1}{10}=0,1$

6. вычислите предел последовательности $lim_{x to 4} frac{x^2-16}{x^2-5x+4}$

$lim_{x to 4} frac{x^2-16}{x^2-5x+4}=lim_{x to 4} frac{(x-4)(x+4)}{x^2-4x-x+4}=lim_{x to 4} frac{(x-4)(x+4)}{x(x-4)-(x-4)}=lim_{x to 4} frac{(x-4)(x+4)}{(x-4)(x -1)}=lim_{x to 4} frac{x+4}{x -1}=frac{4+4}{4-1}=frac{8}{3}=2frac{2}{3}$

7. Вычислите предел функции $lim_{x to 2} frac{sqrt{x-1}-1 }{sqrt{x}-sqrt{2}}$

$lim_{x to 2} frac{sqrt{x-1}-1 }{sqrt{x}-sqrt{2} }= lim_{x to 2} frac{(sqrt{x-1}-1)cdot(sqrt{x-1}+1)cdot(sqrt{x}+sqrt{2}) }{(sqrt{x}-sqrt{2})cdot(sqrt{x}+sqrt{2})cdot(sqrt{x-1}+1)}= lim_{x to 2} frac{(x-1-1)cdot(sqrt{x}+sqrt{2}) }{(x-2)cdot(sqrt{x-1}+1)}= lim_{x to 2} frac{(x-2)cdot(sqrt{x}+sqrt{2}) }{(x-2)cdot(sqrt{x-1}+1)}=lim_{x to 2} frac{sqrt{x}+sqrt{2} }{sqrt{x-1}+1}=frac{sqrt{2}+sqrt{2}}{sqrt{2-1}+1 }=frac{2sqrt{2} }{2}=sqrt{2}$