1. ## 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...

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

And...

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

Then using hospital's rule...

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

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

2. L'Hopital ?

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

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

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

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

$\displaystyle \ln{y} = 1$

limit is $\displaystyle y = e$

3. Oh, I forgot about using that on the second part! Thank you!!!!!

4. 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...

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

And...

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

Then using hospital's rule...

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

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 $\displaystyle x \to \infty$.

$\displaystyle e^x \to \infty$

$\displaystyle x \to \infty$

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

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

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?

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

$\displaystyle e^x \to \infty$

$\displaystyle x \to \infty$

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

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

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?

$\displaystyle \infty^0$ is an indeterminate form, not 1.

6. 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...

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

And...

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

Then using hospital's rule...

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

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