limit as x approaches infinity of x^3sin(1/x)

解答

x \to \infty lim (x^{3} sin (\frac{1}{x}))

\infty

求解步骤

x \to \infty lim (x^{3} sin (\frac{1}{x}))

改写以应用洛必达法则

= x \to \infty lim (\frac{sin ( \frac{1}{x} )}{\frac{1}{x ^{3}}})

使用洛必达法则

= x \to \infty lim \frac{- \frac{c o s ( \frac{1}{x} )}{x ^{2}}}{- \frac{3}{x ^{4}}}

化简

\frac{- \frac{c o s ( \frac{1}{x} )}{x ^{2}}}{- \frac{3}{x ^{4}}} : \frac{1}{3} x^{2} cos (\frac{1}{x})

= x \to \infty lim (\frac{1}{3} x^{2} cos (\frac{1}{x}))

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

= \frac{1}{3} \cdot x \to \infty lim (x^{2} cos (\frac{1}{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

= \frac{1}{3} \cdot x \to \infty lim (x^{2}) \cdot x \to \infty lim (cos (\frac{1}{x}))

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

x \to \infty lim (cos (\frac{1}{x})) = 1

= \frac{1}{3} \cdot \infty \cdot 1

运用无穷大属性:

c \cdot \infty = \infty

= \infty

d er i v a t i v e arctan (6 y^{5} + 1)

\frac{\partial}{\partial x} (y^{2} cos (y x))

\int 2 x x - 1 d x

\frac{d}{d x} (x (3 - 2 x)^{5})

y^{^{''}} + 4 y = t^{2} e^{3 t} + 3