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limit as x approaches 0+of xln(tan(9x))

解答

x \to 0 + lim (x ln (tan (9 x)))

解答

0

求解步骤

x \to 0 + lim (x ln (tan (9 x)))

改写以应用洛必达法则

= x \to 0 + lim (\frac{ln ( tan ( 9 x ) )}{\frac{1}{x}})

使用洛必达法则

= x \to 0 + lim (\frac{9 sec ^{2} ( 9 x ) cot ( 9 x )}{- \frac{1}{x ^{2}}})

化简

= x \to 0 + lim (- 9 x^{2} sec^{2} (9 x) cot (9 x))

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

= - 9 \cdot x \to 0 + lim (x^{2} sec^{2} (9 x) cot (9 x))

用 sin, cos 表示

= - 9 \cdot x \to 0 + lim (\frac{x ^{2}}{cos ( 9 x ) sin ( 9 x )})

\frac{x ^{2}}{cos ( 9 x ) sin ( 9 x )} = \frac{x ^{2}}{sin ( 9 x )} \cdot \frac{1}{cos ( 9 x )}

= - 9 \cdot x \to 0 + lim (\frac{x ^{2}}{sin ( 9 x )} \cdot \frac{1}{cos ( 9 x )})

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

不定型除外

= - 9 \cdot x \to 0 + lim (\frac{x ^{2}}{sin ( 9 x )}) \cdot x \to 0 + lim (\frac{1}{cos ( 9 x )})

x \to 0 + lim (\frac{x ^{2}}{sin ( 9 x )}) = 0

x \to 0 + lim (\frac{1}{cos ( 9 x )}) = 1

= - 9 \cdot 0 \cdot 1

化简

= 0

作图

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