limit as x approaches 0-of 7(1/x-cot(x))

解

x \to 0 - lim (7 (\frac{1}{x} - cot (x)))

0

解答ステップ

x \to 0 - lim (7 (\frac{1}{x} - cot (x)))

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 7 \cdot 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 )}

= 7 \cdot 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

= 7 \cdot \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

= 7 \cdot \frac{\frac{2}{3}}{- \infty}

簡素化

7 \cdot \frac{\frac{2}{3}}{- \infty} : 0

= 0