L'Hospital's rule find limit

can someone tell me what im doing wrong? i always get the limit as 9
lim (e^x + x)^(9/x) as x approaches infinity.
heres my steps:
9/x * ln (e^x + x)
= (ln (e^x + x)) / (x / 9)
= ((1 / (e^x + x)) * e^x + 1) / (1/9)
= 9e^x + 1 / e^x + x
= lim 9e^x / e^x as x approaches infinity = 9

lim (x-->Infinity) (e^x + x)^(9/x)
= lim(x--> Infinity) (9/x)*ln(e^x + x)
= lim(x--> Infinity) (9*ln(e^x + x)/(x)
= lim(x--> Infinity) (9*(e^x + 1)/(e^x+x))/(1)
= lim(x--> Infinity) 9*(e^x)/(e^x+1)
= lim(x--> Infinity) 9*(e^x)/(e^x) = 9

Now, because you placed a natural logarithm into the limit, you must reverse that value at the end after the limit is found; so, we raise it to e, making the ending limit, e^9.

You simply forgot to reverse the natural log after you had originally simplified the equation.
There you go. Hope that helps.

tende to infinite.