数学题,函数的极限lim (x²/x²-1)3x次方 2.lim(tanx)tan2x次方x→无穷大 x→π/4

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数学题,函数的极限lim(x²/x²-1)3x次方2.lim(tanx)tan2x次方x→无穷大x→π/4数学题,函数的极限lim(x²/x²-1)3x次方2.

数学题,函数的极限lim (x²/x²-1)3x次方 2.lim(tanx)tan2x次方x→无穷大 x→π/4
数学题,函数的极限
lim (x²/x²-1)3x次方 2.lim(tanx)tan2x次方
x→无穷大 x→π/4

数学题,函数的极限lim (x²/x²-1)3x次方 2.lim(tanx)tan2x次方x→无穷大 x→π/4
lim(x->∞) [x²/(x²-1)]^3x
=lim(x->∞) [1+1/(x²-1)]^3x
=lim(x->∞) {[1+1/(x²-1)]^(x²-1)}^[(3x)/(x²-1)]
∵lim(x->∞) {[1+1/(x²-1)]^(x²-1)}=e
lim(x->∞) [(3x)/(x²-1)] = 0
=e^0
=1
2.
令:tanx = t ,tan2x = 2t/(1-t²) ,x→π/4 ,t->1
lim( x→π/4) (tanx)^(tan2x)
=lim( t→1) (t)^[2t/(1-t²)]
=lim( t→1) [1+(t-1)]^{ 1/(t-1) * [2t(t-1)/(1-t²)] }
=lim( t→1) {[1+(t-1)]^{ 1/(t-1)}^[-2t/(1+t)]
∵lim( t→1) {[1+(t-1)]^{ 1/(t-1)} = e
lim( t→1) [-2t/(1+t)] = -1
= e^(-1)
= 1/e