Find limit of ((x² + x + 1)/?(x^4 + x² + 1))^x as x -> infinity?

[scythian1950]

This was inspired by a pretty Y!A question posted recently.

 

[ksoileau]

(x² + x + 1)/?(x^4 + x² + 1)=(x² + x + 1)/x^2/?(1 + 1/x² + 1/x^4)
=(1 + 1/x + 1/x^2)/?(1 + 1/x² + 1/x^4)

1/?(1 + 1/x² + 1/x^4)=1-1/2/x^2+O(1/x^4)

so
(1 + 1/x + 1/x^2)/?(1 + 1/x² + 1/x^4)
=(1 + 1/x + 1/x^2)*(1-1/2/x^2+O(1/x^4))
=1+1/x+O(1/x^2)

thus
((x² + x + 1)/?(x^4 + x² + 1))^x
=(1+1/x+O(1/x^2))^x
=(1+1/x)^x+O(1/x)→e