You and I must have been working on this problem at the same time. I did it differently
=lim ex(1+(x/ex )]9/x
=lim ex9/x(1+(x/ex )]9/x
=lim e9(1+(x/ex )]9/x
Extra Little Note:
SlipEternal turned it into a algebraic fraction so she/he could use L'hopital's Rule to solve it.
L'H˘pital's rule - Wikipedia, the free encyclopedia
To recap, is not defined because is not a number. So, instead, I altered the form so that I could apply L'Hospital's Rule.
I have a slightly different explanation of why you can't just substitute in infinity.
ex+x is infinitely large as x approaches infinity.
9/x is infinitely small as x approaches infinity.
They are not approaching infinity or 0 at the same rate. You have to change the form or it will not be valid.
Using your logic I think you might be tempted to say that
lim [ex]9/x = 1
but you can see that it can be transformed into
Think it over, I hope that it will help.