设f(x)在x=0的邻域内有三阶导数,且x->0时,lim(1+x+f(x)/x)^(1/x)=e^3.求(1):f(0),f'(0).f''(0)求(2)lim(1+f(x)/x)^(1/x).
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![设f(x)在x=0的邻域内有三阶导数,且x->0时,lim(1+x+f(x)/x)^(1/x)=e^3.求(1):f(0),f'(0).f''(0)求(2)lim(1+f(x)/x)^(1/x).](/uploads/image/z/1983266-26-6.jpg?t=%E8%AE%BEf%28x%29%E5%9C%A8x%3D0%E7%9A%84%E9%82%BB%E5%9F%9F%E5%86%85%E6%9C%89%E4%B8%89%E9%98%B6%E5%AF%BC%E6%95%B0%2C%E4%B8%94x-%3E0%E6%97%B6%2Clim%281%2Bx%2Bf%28x%29%2Fx%29%5E%281%2Fx%29%3De%5E3.%E6%B1%82%EF%BC%881%EF%BC%89%EF%BC%9Af%280%29%2Cf%27%280%29.f%27%27%280%29%E6%B1%82%EF%BC%882%EF%BC%89lim%281%2Bf%28x%29%2Fx%29%5E%281%2Fx%29.)
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设f(x)在x=0的邻域内有三阶导数,且x->0时,lim(1+x+f(x)/x)^(1/x)=e^3.求(1):f(0),f'(0).f''(0)求(2)lim(1+f(x)/x)^(1/x).
设f(x)在x=0的邻域内有三阶导数,且x->0时,lim(1+x+f(x)/x)^(1/x)=e^3.求(1):f(0),f'(0).f''(0)
求(2)lim(1+f(x)/x)^(1/x).
设f(x)在x=0的邻域内有三阶导数,且x->0时,lim(1+x+f(x)/x)^(1/x)=e^3.求(1):f(0),f'(0).f''(0)求(2)lim(1+f(x)/x)^(1/x).
1、条件等价于ln(1+x+f(x)/x)/x的极限是3.显然f(0)=0.另外,由于分母x趋于0,因此分子ln函数必趋于0,于是得x+f(x)/x趋于0,f(x)/x趋于0,即[f(x)-f(0)]/(x-0)趋于0,于是f'(0)=0.
利用Taylor展式得f(x)/x=f''(0)x^2/2+小o(x^2),因此
ln(1+x+f(x)/x^2)等价于x+f''(0)x/2,故极限值为1+f''(0)/2=3,解得f''(0)=4.
2、lim ln(1+f(x)/x)/x=lim ln(1+f''(0)x/2+小o(x))/x=lim 【f''(0)x/2+小o(x)】/x=f''(0)/2=2.
因此原极限为e^2
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