limit as x approaches 0+of 1/x-cot(x)

解

x \to 0 + lim (\frac{1}{x} - cot (x))

0

解答ステップ

x \to 0 + lim (\frac{1}{x} - cot (x))

\frac{1}{x} - cot (x)

の共役で乗じる

\frac{\frac{1}{x ^{2}} - cot ^{2} ( x )}{\frac{1}{x} + cot ( x )}

= x \to 0 + lim (\frac{\frac{1}{x ^{2}} - cot ^{2} ( x )}{\frac{1}{x} + cot ( x )})

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 _{x \to 0 +} ( \frac{1}{x ^{2}} - cot ^{2} ( x ) )}{lim _{x \to 0 +} ( \frac{1}{x} + cot ( x ) )}

x \to 0 + lim (\frac{1}{x ^{2}} - cot^{2} (x)) = \frac{2}{3}

x \to 0 + lim (\frac{1}{x} + cot (x)) = \infty

= \frac{\frac{2}{3}}{\infty}

無限大の特性を適用する:

\frac{c}{\infty} = 0

= 0