limit as n approaches infinity of (nsqrt(n)-1)/(n^2+1)

解

n \to \infty lim (\frac{n n - 1}{n ^{2} + 1})

0

解答ステップ

n \to \infty lim (\frac{n n - 1}{n ^{2} + 1})

最高の分母べき乗で割る:

\frac{\frac{1}{n} - \frac{1}{n ^{2}}}{1 + \frac{1}{n ^{2}}}

= n \to \infty lim (\frac{\frac{1}{n} - \frac{1}{n ^{2}}}{1 + \frac{1}{n ^{2}}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

不定形式の場合を除く

= \frac{lim _{n \to \infty} ( \frac{1}{n} - \frac{1}{n ^{2}} )}{lim _{n \to \infty} ( 1 + \frac{1}{n ^{2}} )}

n \to \infty lim (\frac{1}{n} - \frac{1}{n ^{2}}) = 0

n \to \infty lim (1 + \frac{1}{n ^{2}}) = 1

= \frac{0}{1}

規則を適用

\frac{0}{a} = 0, a \neq = 0

= 0