高数极限3道题1题limx→正无穷(1 - 2/3x)^4x+12题limx→正无穷(x/1+x)^-2x+13题limx→正无穷(x+1/x-1)^x2题是(x除1+x)^-2x+1

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高数极限3道题1题limx→正无穷(1-2/3x)^4x+12题limx→正无穷(x/1+x)^-2x+13题limx→正无穷(x+1/x-1)^x2题是(x除1+x)^-2x+1高数极限3道题1题l

高数极限3道题1题limx→正无穷(1 - 2/3x)^4x+12题limx→正无穷(x/1+x)^-2x+13题limx→正无穷(x+1/x-1)^x2题是(x除1+x)^-2x+1
高数极限3道题
1题limx→正无穷(1 - 2/3x)^4x+1
2题limx→正无穷(x/1+x)^-2x+1
3题limx→正无穷(x+1/x-1)^x
2题是(x除1+x)^-2x+1

高数极限3道题1题limx→正无穷(1 - 2/3x)^4x+12题limx→正无穷(x/1+x)^-2x+13题limx→正无穷(x+1/x-1)^x2题是(x除1+x)^-2x+1
1题.原式=lim(x->+∞){[1-2/(3x)]^(4x+1)}
=lim(x->+∞){[1+(-2)/(3x)]^[(3x)/(-2)]}^[(-2)(4x+1)/(3x)]}
=e^{lim(x->+∞){[(-2)(4x+1)/(3x)]} (利用特殊极限)
=e^(-8/3)
2题.原式=lim(x->+∞){[1+(-1)/(1+x)]^(-2x+1)}
=lim(x->+∞){[1+(-1)/(1+x)]^[(1+x)/(-1)]}^[(2x-1)/(1+x)]
=e^{lim(x->+∞)[(2x-1)/(1+x)]} (利用特殊极限)
=e^2
3题.原式=lim(x->+∞){[(x+1)/(x-1)]^x}
=lim(x->+∞){[1+2/(x-1)]^[(x-1)/2]}^[2x/(x-1)]
=e^{lim(x->+∞)[2x/(x-1)]} (利用特殊极限)
=e^2
注:利用特殊极限lim(x->+∞)[(1+1/x)^x]=1

同学你给的题目叫别人怎么做啊, 第二题是(1/x+x)^(-2x+1)
吧,注意该加括号的地方要加:(x+1/(x-1))^x,(1 - 2/3x)^(4x+1), 要不然别人怎么做啊

1.
lim(1 - 2/3x)^(4x+1)
=lim(1-2/3x)^[(-3x/2)*(-2/3x)*(4x+1)]
=lim[(1-2/3x)^(-3x/2)]^[-2(4x+1)/3x]
lim(1-2/3x)^(-3x/2)=e,lim[-2(4x+1)/3x]=-8/3

lim[(1-2/3x)^(-3x/2)]^[-2(4x+1)...

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1.
lim(1 - 2/3x)^(4x+1)
=lim(1-2/3x)^[(-3x/2)*(-2/3x)*(4x+1)]
=lim[(1-2/3x)^(-3x/2)]^[-2(4x+1)/3x]
lim(1-2/3x)^(-3x/2)=e,lim[-2(4x+1)/3x]=-8/3

lim[(1-2/3x)^(-3x/2)]^[-2(4x+1)/3x]=e^(-8/3)
2.
lim(x/1+x)^(-2x+1)
=lim[(x+1-1)/(x+1)]^(-2x+1)
=lim[1-1/(x+1)]^[(-(x+1))*(-1/(x+1))*(-2x+1)]
=lim{[1-1/(x+1)]^(-(x+1))}^[(2x-1)/(x+1)]
lim[1-1/(x+1)]^(-(x+1))=e,lim(2x-1)/(x+1)=2

lim{[1-1/(x+1)]^(-(x+1))}^[(2x-1)/(x+1)]=e^2
3.
lim(x+1/x-1)^x
=lim[(x-1+2)/(x-1)]^x
=lim[1+2/(x-1)]^[((x-1)/2)*(2/(x-1))*x]
=lim{[1+2/(x-1)]^((x-1)/2)}^[2x/(x-1)]
lim[1+2/(x-1)]^((x-1)/2)=e,lim2x/(x-1)=2

lim{[1+2/(x-1)]^((x-1)/2)}^[2x/(x-1)]=e^2

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