x趋于无穷,(x^2+a^2/x^2-a^2)^(x^2)的极限

题目

x趋于无穷,(x^2+a^2/x^2-a^2)^(x^2)的极限

答案

令m=x²
则m→∞
原式=[(m-a²+2a²)/(m-a²)]^m
=[1+2a²/(m-a²)]^m
令2a²/(m-a²)=1/n
则m→∞有n→∞
m=2na²+a²
原式=(1+1/n)^(2na²+a²)
=(1+1/n)^(2na²)*(1+1/n)^a²
=[(1+1/n)^n]^2a²*(1+1/n)^a²
n→∞,a是常数,所以(1+1/n)^a²极限是1
(1+1/n)^n极限是e
所以原来极限=e^(2a²)