# Limit question

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• Feb 18th 2009, 06:27 PM
cielroi
Limit question
Thanks for the help.

This is a limit question that is the last on the list that I can't just figure out.

lim x -> infinity

(e^x + x) ^ (1/x)

I'm sorry for the lack of formatting. When I applied math tags, it didn't come out right.

I got as far as...

$ln(y) = (1/x) * ln(e^x + x)$

And...

$ln(y) = ln(e^x + x)/x$

Then using hospital's rule...

lim x -> inf $(1+e^x)/(x+e^x)$

What is the limit above? That looks like the only problem that's keeping me down.
• Feb 18th 2009, 06:43 PM
skeeter
L'Hopital ?

$\lim_{x \to \infty} \frac{\ln(e^x + x)}{x} =$

$\lim_{x \to \infty} \frac{e^x + 1}{e^x + x} =$

$\lim_{x \to \infty} \frac{e^x}{e^x + 1} =$

$\lim_{x \to \infty} \frac{e^x}{e^x} = 1$

$\ln{y} = 1$

limit is $y = e$
• Feb 18th 2009, 06:51 PM
cielroi
Oh, I forgot about using that on the second part! Thank you!!!!!
• Feb 18th 2009, 06:51 PM
Prove It
Quote:

Originally Posted by cielroi
Thanks for the help.

This is a limit question that is the last on the list that I can't just figure out.

lim x -> infinity

(e^x + x) ^ (1/x)

I'm sorry for the lack of formatting. When I applied math tags, it didn't come out right.

I got as far as...

$ln(y) = (1/x) * ln(e^x + x)$

And...

$ln(y) = ln(e^x + x)/x$

Then using hospital's rule...

lim x -> inf $(1+e^x)/(x+e^x)$

What is the limit above? That looks like the only problem that's keeping me down.

This is a bit messy, but I look at what happens to each part as $x \to \infty$.

$e^x \to \infty$

$x \to \infty$

So $e^x + x \to \infty$.

$\frac{1}{x} \to 0$.

So we have a very big number taken to the power of a very small number (in this case, 0).

What is anything to the power of 0?
• Feb 18th 2009, 06:55 PM
skeeter
Quote:

Originally Posted by Prove It
This is a bit messy, but I look at what happens to each part as $x \to \infty$.

$e^x \to \infty$

$x \to \infty$

So $e^x + x \to \infty$.

$\frac{1}{x} \to 0$.

So we have a very big number taken to the power of a very small number (in this case, 0).

What is anything to the power of 0?

$\infty^0$ is an indeterminate form, not 1.
• Feb 18th 2009, 06:56 PM
mollymcf2009
Quote:

Originally Posted by cielroi
Thanks for the help.

This is a limit question that is the last on the list that I can't just figure out.

lim x -> infinity

(e^x + x) ^ (1/x)

I'm sorry for the lack of formatting. When I applied math tags, it didn't come out right.

I got as far as...

$ln(y) = (1/x) * ln(e^x + x)$

And...

$ln(y) = ln(e^x + x)/x$

Then using hospital's rule...

lim x -> inf $(1+e^x)/(x+e^x)$

What is the limit above? That looks like the only problem that's keeping me down.

You need to l'Hopital again, because you have $\frac{\infty}{\infty}$. Then it looks like you will have to do it maybe two more times, which I think will give you $\frac{1}{1} = 1$ So your limit will be $e^1$ which = e