Пределы: http://mathprofi.ru/predely_primery_reshenii.html
[tex]\lim_{n \to \infty} (\sqrt{n^2-20n+1} -\sqrt{n^2+1}) = \infty-\infty=\\\\\\= \lim_{n \to \infty} \frac{(\sqrt{n^2-20n+1} -\sqrt{n^2+1})(\sqrt{n^2-20n+1} +\sqrt{n^2+1})}{(\sqrt{n^2-20n+1} +\sqrt{n^2+1})} =\\\\\\=\lim_{n \to \infty} \frac{n^2-20n+1 +n^2+1}{(\sqrt{n^2-20n+1} +\sqrt{n^2+1})} =\\\\\\=\lim_{n \to \infty} \frac{-20n}{\sqrt{n^2-20n+1} +\sqrt{n^2+1}}=\lim_{n \to \infty} \frac{-20n/n }{\sqrt{n^2/n^2 -20n/n^2 +1/n^2}+\sqrt{n^2/n^2+1/n^2}}=\\=\frac{-20}{\sqrt{1-0+0}+\sqrt{1+0} } = -10[/tex]
2)[tex]\lim_{n \to \infty} (\frac{n^2+1}{n^2} )^5^n= \frac{\infty}{\infty} = \lim_{n \to \infty} \frac{n^2^*^5^n}{n^2^*^5^n} = lim_{n \to \infty} \frac{n^1^0^n}{n^1^0^n}=\\\\\=lim_{n \to \infty} \frac{n^1^0^n/n^1^0^n}{n^1^0^n/n^1^0^n}= \frac{1}{1} =1[/tex]
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Пределы: http://mathprofi.ru/predely_primery_reshenii.html
[tex]\lim_{n \to \infty} (\sqrt{n^2-20n+1} -\sqrt{n^2+1}) = \infty-\infty=\\\\\\= \lim_{n \to \infty} \frac{(\sqrt{n^2-20n+1} -\sqrt{n^2+1})(\sqrt{n^2-20n+1} +\sqrt{n^2+1})}{(\sqrt{n^2-20n+1} +\sqrt{n^2+1})} =\\\\\\=\lim_{n \to \infty} \frac{n^2-20n+1 +n^2+1}{(\sqrt{n^2-20n+1} +\sqrt{n^2+1})} =\\\\\\=\lim_{n \to \infty} \frac{-20n}{\sqrt{n^2-20n+1} +\sqrt{n^2+1}}=\lim_{n \to \infty} \frac{-20n/n }{\sqrt{n^2/n^2 -20n/n^2 +1/n^2}+\sqrt{n^2/n^2+1/n^2}}=\\=\frac{-20}{\sqrt{1-0+0}+\sqrt{1+0} } = -10[/tex]
2)[tex]\lim_{n \to \infty} (\frac{n^2+1}{n^2} )^5^n= \frac{\infty}{\infty} = \lim_{n \to \infty} \frac{n^2^*^5^n}{n^2^*^5^n} = lim_{n \to \infty} \frac{n^1^0^n}{n^1^0^n}=\\\\\=lim_{n \to \infty} \frac{n^1^0^n/n^1^0^n}{n^1^0^n/n^1^0^n}= \frac{1}{1} =1[/tex]