# Thread: Limits

1. ## Limits

Hey!

I'm not quite sure how to proceed in calculating this limit.
If you could look at it and give me some feedback I would appreciate it.

$\displaystyle \lim_{x \to +\infty} \frac{xln(e^x+e^{-x})+\sqrt{x^3+1}}{x^2+1}$

Thanks in advance.

2. these might help you

$\displaystyle xln(e^x+e^{-x})$

=$\displaystyle x[ln(1+e^{-2x})+x]$

3. After using that, divide both numerator and denominator by $\displaystyle x^2$.

4. Thank you!

5. Here's a hand-waving argument, for large $\displaystyle x$

$\displaystyle \frac{x \ln(e^x+e^{-x})+\sqrt{x^3+1}}{x^2+1} \approx \frac{x \ln(e^x)}{x^2} = \frac{x^2}{x^2} = 1$

6. Originally Posted by mei
Hey!

I'm not quite sure how to proceed in calculating this limit.
If you could look at it and give me some feedback I would appreciate it.

$\displaystyle \lim_{x \to +\infty} \frac{xln(e^x+e^{-x})+\sqrt{x^3+1}}{x^2+1}$

Thanks in advance.
You can divide all three terms by $\displaystyle x^2$

and concentrate on $\displaystyle {\ln(e^x+e^{-x})\over x}$