limit as x approaches 0 of 2x^2*cot^2(x)

解答

x \to 0 lim (2 x^{2} \cdot cot^{2} (x))

2

求解步骤

x \to 0 lim (2 x^{2} cot^{2} (x))

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

= 2 \cdot x \to 0 lim (x^{2} cot^{2} (x))

使用指数法则:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

x^{2} cot^{2} (x) = (x cot (x))^{2}

= 2 \cdot x \to 0 lim ((x cot (x))^{2})

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

不定型除外

= 2 (x \to 0 lim (x cot (x)))^{2}

改写以应用洛必达法则

= 2 (x \to 0 lim (\frac{x}{\frac{1}{c o t ( x )}}))^{2}

使用洛必达法则

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

代入值

x = 0

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

化简

2 (\frac{1}{sec ^{2} ( 0 )})^{2} : 2

= 2

x \to \frac{π}{2} lim (sin (x) + 1)

x \to 1 lim (\frac{∣ x - 3 ∣}{x - 3})

x \to 0 lim (∣ sin (x) ∣ - \frac{1}{2})

x \to 8 lim (\frac{3}{10})

x \to 0 lim (3 x \cdot cot (3 x))