limit as x approaches 0+of xcsc^2(x)

解答

x \to 0 + lim (x csc^{2} (x))

\infty

求解步骤

x \to 0 + lim (x csc^{2} (x))

用 sin, cos 表示

= x \to 0 + lim (\frac{x}{sin ^{2} ( x )})

使用洛必达法则

= x \to 0 + lim (\frac{1}{sin ( 2 x )})

x

从右侧趋近于

0, x > 0 \Rightarrow sin (2 x) > 0

分母从右侧趋近于 0，为正值

= \infty

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x \to \infty lim ((\frac{2 ln ( x )}{ln ( 7 x ^{2} )})^{l n (x)})

x \to - 6 lim (\frac{4}{x + 6})