Question

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

Answer 1

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