Lim (e^(x)+x)^(1/x)
x->∞
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.
Ok great! My bad, I didnt see what you were doing, so as you noticed
$\displaystyle \lim_{x\to\infty}\frac{\ln\left(e^x+x\right)}{x}=\ lim_{x\to\infty}\frac{\frac{e^x+1}{e^x+x}}{1}=1$
And also as you stated, remebering that $\displaystyle L$ is our orginal limit if
$\displaystyle \ln(L)=1\Rightarrow{L=e^1=e}$
Ok, let $\displaystyle f(x)=(e^x+x)^{\frac{1}{x}}$
My 89 gives $\displaystyle f(230)=2.71828$
$\displaystyle f(2300)=2.71828$
$\displaystyle f(2300000)=2.71828$
but
$\displaystyle f(23000000)=\infty$
I assume that the caclulator does not have enough memory to do this, so it assumes it is zero.
And please, for your sake, don't rely on your calculator to do limits.
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.
Also be weary for, some limits converge very slowly
So $\displaystyle f\left(10^{10^{10}}\right)$ might not be close to $\displaystyle \lim_{x\to\infty}f(x)$
Also note that this will method of plug and chug will give you an approximation, so the answer might be $\displaystyle \sqrt{2}\pi$ but you would never guess that from your evaluation giving you $\displaystyle 4.4428$
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.