limit as x approaches infinity of x(arctan(x)-pi/2)

解答

x \to \infty lim (x (arctan (x) - \frac{π}{2}))

- 1

求解步骤

x \to \infty lim (x (arctan (x) - \frac{π}{2}))

改写以应用洛必达法则

= x \to \infty lim (\frac{arctan ( x ) - \frac{π}{2}}{\frac{1}{x}})

使用洛必达法则

= x \to \infty lim (\frac{\frac{1}{x ^{2} + 1}}{- \frac{1}{x ^{2}}})

化简

\frac{\frac{1}{x ^{2} + 1}}{- \frac{1}{x ^{2}}} : - \frac{x ^{2}}{x ^{2} + 1}

= x \to \infty lim (- \frac{x ^{2}}{x ^{2} + 1})

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

= - x \to \infty lim (\frac{x ^{2}}{x ^{2} + 1})

除以分母的最高次幂:

\frac{1}{1 + \frac{1}{x ^{2}}}

= - x \to \infty lim (\frac{1}{1 + \frac{1}{x ^{2}}})

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

不定型除外

= - \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( 1 + \frac{1}{x ^{2}} )}

x \to \infty lim (1) = 1

x \to \infty lim (1 + \frac{1}{x ^{2}}) = 1

= - \frac{1}{1}

化简

= - 1

d er i v a t i v e f (x) = (7 x^{6} + 8 x^{3})^{4}

4 y^{^{''}} - 4 y^{^{'}} - 3 y = 0, y (0) = 1, y^{^{'}} (0) = 5

x \to 0 + lim ((cot (x))^{s i n (x)})

x^{2} \frac{d y}{d x} - 2 x y = 3 y^{4}

\int \frac{5}{x} d x