limit as x approaches 0 of x/(tan(5x))

解答

x \to 0 lim (\frac{x}{tan ( 5 x )})

\frac{1}{5}

十进制

0.2

求解步骤

x \to 0 lim (\frac{x}{tan ( 5 x )})

用 sin, cos 表示

= x \to 0 lim (\frac{x cos ( 5 x )}{sin ( 5 x )})

使用基本三角恒等式:

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

= x \to 0 lim (x cot (5 x))

改写以应用洛必达法则

= x \to 0 lim (\frac{x}{\frac{1}{c o t ( 5 x )}})

使用洛必达法则

= x \to 0 lim (\frac{1}{sec ^{2} ( 5 x ) \cdot 5})

代入值

x = 0

= \frac{1}{sec ^{2} ( 5 \cdot 0 ) \cdot 5}

sec^{2} (5 \cdot 0) \cdot 5 = 5

= \frac{1}{5}

\int \frac{sin ( 2 x )}{cos ^{2} ( 2 x )} d x

x \to 1 lim (x + 2)

\frac{d y}{d x} = \frac{y}{x} + 1

y^{^{'}} + 4 x y = x e^{x^{2}}, y (0) = 1

\frac{\partial}{\partial x} (ln (5 x^{2} + 8 y^{2} + 4))