limit as t approaches 0 of t/(tan(3t))

解答

t \to 0 lim (\frac{t}{tan ( 3 t )})

\frac{1}{3}

十进制

0.33333 \dots

求解步骤

t \to 0 lim (\frac{t}{tan ( 3 t )})

用 sin, cos 表示

= t \to 0 lim (\frac{t cos ( 3 t )}{sin ( 3 t )})

使用基本三角恒等式:

\frac{cos ( x )}{sin ( x )} = cot (x)

= t \to 0 lim (t cot (3 t))

改写以应用洛必达法则

= t \to 0 lim (\frac{t}{\frac{1}{c o t ( 3 t )}})

使用洛必达法则

= t \to 0 lim (\frac{1}{sec ^{2} ( 3 t ) \cdot 3})

代入值

t = 0

= \frac{1}{sec ^{2} ( 3 \cdot 0 ) \cdot 3}

sec^{2} (3 \cdot 0) \cdot 3 = 3

= \frac{1}{3}

\frac{d}{d x} (arcsin (\frac{x}{6}))

\frac{d}{d x} (e^{x y})

\frac{d}{d x} (csc (x) - cot (x))

\int (18 x + 4) (9 x^{2} + 4 x)^{5} d x

\frac{\partial}{\partial x} (- 4 y^{5} sin (4 x))