What I came to was e^(1). I'm not sure if this is the right answer, the TI-83 has it at e^(1) up to 230, then at 231 it's undefined. I don't understand why, my TI-89 has it as ∞ past 231.
Sorry about the formatting, I couldn't get the formatting to work, tried copy pasting math codes but it came out in error. Anyway, here goes:
e^ln(e^x+x)^(1/x) = e^(e^x+1)/(e^x+x) = e^(e^x/(e^x+1)) = e^(1), as x approaches infinity. The 83 gives undefined as x is greater than 230 and 89 has it as infinity, but at 230 it's e^(1). It doesn't make sense to me.
Well that is very odd. I was completely sure e^(1) was the answer algebraically, I just couldn't figure out why the calculator was trying to throw me off. I use the calculator as a "check" to make sure I have the right answer, but this case it didn't give me the right answer. Still a bit puzzled as to why this problem will do that, since as x-> infinity, y is approaching a constant value eh? Maybe it's the e that is throwing it off? Most other problems tend to give the right answer for horizontal limits.
What I do is, if I do it algebraically and get sqrt2 pi, I will plug the equation into my calculator and see what it gives me. If it gives me 4.4428, I will approximate sqrt2 pi into my calculator to see if it gives me 4.4428, if it does, then I may have the right answer. Is there a reason I would want to use 10^10^10 as x? I usually just use 999 or anything random.