Question

求几题微积分计算

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Answer 1

(1)
lim(x->0+) sinx.lnx
=lim(x->0+) lnx/cscx (∞/∞分子分母分别求导)
=lim(x->0+) (1/x)/[-(cotx)^2]
=lim(x->0+) -(tanx)^2/x
=0
(2)
lim(x->0) [ 1/x^2 - 1/(xtanx) ]
=lim(x->0) (tanx - x)/(x^2.tanx)
=lim(x->0) (tanx - x)/x^3 (0/0分子分母分别求导)
=lim(x->0) [(secx)^2 - 1 ]/(3x^2)
=lim(x->0) (tanx)^2/(3x^2)
=lim(x->0) x^2/(3x^2)
=1/3
(3)
lim(x->+∞) [ √(x+√x) -√x ]
=lim(x->+∞) [ (x+√x) -x ] / [ √(x+√x) +√x ]
=lim(x->+∞) √x / [ √(x+√x) +√x ]
分子分母同时除以√x
=lim(x->+∞) 1 / [ √(1+1/√x) +1 ]
=1/2
(1)
lim(x->0) (e^x -x-1)/[√(1+x^2) -1]
=lim(x->0) (e^x -x-1).[√(1+x^2) +1]/[(1+x^2) -1]
=2lim(x->0) (e^x -x-1)/x^2 (0/0 分子分母分别求导)
=2lim(x->0) (e^x -1)/(2x)
=2lim(x->0) x/(2x)
=1
(2)
lim(x->0) [√(1+x) -√(1+sinx)]/arctan(x^3)
=lim(x->0) [(1+x) -(1+sinx)]/{ arctan(x^3) . [√(1+x) +√(1+sinx)] }
=(1/2)lim(x->0) (x-sinx)/arctan(x^3)
=(1/2)lim(x->0) (x-sinx)/ (x^3) (0/0 分子分母分别求导)
=(1/2)lim(x->0) (1-cosx)/ (3x^2)
=(1/2)lim(x->0) (1/2)x^2/ (3x^2)
=1/12
(3)
lim(x->+∞) x^100/e^x =0
(4)
lim(x->+∞) [2^x +x^100-(lnx)^1000] /[ 5^x+x^100+3(lnx)^102 ]
=lim(x->+∞) 2^x / 5^x
=lim(x->+∞) (2 /5)^x
=0

Answer 2

方法如下图所示，
请认真查看，
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