e^(-1/x^2)/x x趋向于0 求极限

设y = 1/x²，x = ±y^(-1/2)

e^(-1/x^2)/x
= ±e^(-y) / y^(-1/2)
= ±y^(1/2) / e^y

x → 0 等价于 y → ∞

lim[(e^(-1/x^2))/x, x → 0]
= lim[ ±y^(1/2) / e^y, y → ∞ ]

y^(1/2) / e^y 为 ∞/∞ 型，可用洛必达法则
y^(1/2)求导为(1/2)y^(-1/2)，e^y求导为e^y

lim[(e^(-1/x^2))/x, x → 0]
= lim[ ±y^(1/2) / e^y, y → ∞ ]
= lim[ ±(1/2)y^(-1/2) / e^y, y → ∞ ]
= lim[ ±1 / 2y^(1/2)e^y, y → ∞ ]
= 0

令t = 1/x，原式变为
t e^(-t^2) = t/e^(t^2) , t趋向于无穷
罗比大法则，分子分母求导
lim t/e^(t^2) = 1/(2t*e^(t^2) = 0, t趋向于无穷