Calculating Limits: The Uncertainty of 0*Infinity

[Dell]

in a calculus problem i am doing, i need to find the Limit of

Lim (1+x)ln(e-1/x^2)-x
x->infinity

i have brought it to a point where i have ln(e+1/x^2)+x(ln(e+1/x^2)-1)
therefore my limit is 1-(0*inf)

can i leave this as 1-0=1 and that would be my answer or do i somehow need to change it around to get 0/0 of inf/inf??

 

[MathematicalPhysicist]

I am not sure how did you arrive on your second sentence, but from there, you need to use L'hopital Rule on (ln(e+1/x^2)-1)/(1/x), though if you change variables you get that this is definition of the derivative of the function ln(e+u) at the point u=0, have I ruined it for you, have I? :-)