lim x tends to infinity [(x^2+5x+3)/(x^2+x+2)]^x=?

lim x tends to infinity [(x^2+5x+3)/(x^2+x+2)]^x
=e^lim x tends to infinity x.[(x^2+5x+3)/(x^2+x+2)-1]
=e^lim x tends to infinity x.(4x+1)/(x^2+x+2)
=e^lim x tends to infinity 4x^2(1+1/x)/x^2(1+1/x+2/x^2)
=e^lim x tends to infinity4(1+1/x)/(1+1/x+2/x^2)
applying limit, we get
e^4

Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty