作业帮求导,求极限1.已知f(x)= x^2sin(1/x) x不等于0时, f(x)=0 x=0时. 求f'(x)2.求极限：lim(x->0)[(e^x-1-x)^2/tan(sinx)^2]
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![求导,求极限1.已知f(x)= x^2sin(1/x) x不等于0时, f(x)=0 x=0时. 求f'(x)2.求极限：lim(x->0)[(e^x-1-x)^2/tan(sinx)^2]](/uploads/image/z/5533119-63-9.jpg?t=%E6%B1%82%E5%AF%BC%2C%E6%B1%82%E6%9E%81%E9%99%901.%E5%B7%B2%E7%9F%A5f%28x%29%3D+x%5E2sin%281%2Fx%29++x%E4%B8%8D%E7%AD%89%E4%BA%8E0%E6%97%B6%2C+f%28x%29%3D0+++x%3D0%E6%97%B6.+%E6%B1%82f%27%28x%292.%E6%B1%82%E6%9E%81%E9%99%90%EF%BC%9Alim%28x-%3E0%29%5B%28e%5Ex-1-x%29%5E2%2Ftan%28sinx%29%5E2%5D)
求导,求极限1.已知f(x)= x^2sin(1/x) x不等于0时, f(x)=0 x=0时. 求f'(x)2.求极限：lim(x->0)[(e^x-1-x)^2/tan(sinx)^2]
求导,求极限
1.已知f(x)= x^2sin(1/x) x不等于0时, f(x)=0 x=0时. 求f'(x)
2.求极限：lim(x->0)[(e^x-1-x)^2/tan(sinx)^2]
求导,求极限1.已知f(x)= x^2sin(1/x) x不等于0时, f(x)=0 x=0时. 求f'(x)2.求极限：lim(x->0)[(e^x-1-x)^2/tan(sinx)^2]
1.已知f(x)= x^2sin(1/x) x不等于0时,f(x)=0 x=0时.求f'(x)
x不等于0时,f'(x) = 2xsin(1/x) + x^2cos(1/x)*(-1/x^2)
= 2xsin(1/x) - cos(1/x)
x = 0 时,
lim_{x->0} [f(x)] = lim_{x->0}[x^2sin(1/x)] = 0 = f(0),
所以,f(x)在 x = 0处连续.
lim_{x->0}{[f(x) - f(0)]/(x-0)} = lim_{x->0}[x^2sin(1/x)/x]
= lim_{x->0}[xsin(1/x)] = 0
所以,f(x)在 x = 0处可导,f'(0) = 0.
综合,有,
x不等于0时,f'(x) = 2xsin(1/x) - cos(1/x)
x = 0 时,f'(0) = 0.
2.求极限：lim(x->0)[(e^x-1-x)^2/tan(sinx)^2]
lim_{x->0}[(e^x-1-x)^2/tan(sinx)^2]
= lim_{x->0}[(e^x-1-x)^2/(sinx)^2][(sinx)^2/tan(sinx)^2]
= lim_{x->0}[(e^x-1-x)^2/(sinx)^2]
= lim_{x->0}[(e^x-1-x)^2/x^2] [x^2/(sinx)^2]
= lim_{x->0}[(e^x-1-x)^2/x^2]
= lim_{x->0}[2(e^x-1-x)(e^x - 1)/(2x)]
= lim_{x->0}[e^x-1-x][(e^x - 1)/x]
= lim_{x->0}[e^x-1-x]
= 1 - 1 - 0
= 0.
1 题目看不懂
2 tan(sinx)^2 ~ (sinx)^2 ~ x^2
e^x-1 ~x
用以上等价无穷小代换即可解得