Question

e^(-1/x^2)/x x趋向于0 求极限

Answer 1

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