# Thread: Limit of a function to the power infinite

1. ## Limit of a function to the power infinite

What exactly is ?

(a) $\displaystyle e$
(b) $\displaystyle e^2$
(c) $\displaystyle e^3$
(d) $\displaystyle \frac{1}{e}$

2. ## Re: Limit of a function to the power infinite

$\displaystyle \lim _{x \rightarrow 0} e^{\ln \left( \tan \left( \frac{\pi}{4}+x \right)\right)^\frac{1}{x}$

since e is continuous
$\displaystyle e^{\lim _{x \rightarrow 0}\ln \left( \tan \left( \frac{\pi}{4}+x \right)\right)^\frac{1}{x}$

$\displaystyle \lim _{x \rightarrow 0}\ln \left( \tan \left( \frac{\pi}{4}+x \right)\right)^\frac{1}{x}$

$\displaystyle \lim _{x \rightarrow 0}\left(\frac{1}{x}\right)\ln \left( \tan \left( \frac{\pi}{4}+x \right)\right)$

$\displaystyle \lim _{x \rightarrow 0}\left(\frac{\ln \left( \tan \left( \frac{\pi}{4}+x \right)\right)}{x}\right)$
0/0
use lopital rule

3. ## Re: Limit of a function to the power infinite

Thanks. Some problems though...

How exactly do you make e the base in the second step? (after "since e is continuous")?

4. ## Re: Limit of a function to the power infinite

Originally Posted by cosmonavt
Thanks. Some problems though...

How exactly do you make e the base in the second step? (after "since e is continuous")?
In general, $\displaystyle u=e^{\ln(u)}$, because $\displaystyle y=\ln(u) \Leftrightarrow e^{y}=u \Leftrightarrow e^{\ln(u)}=u$. So for this limit, let $\displaystyle u=\tan\left(\frac{\pi}{4}+x\right)^{\frac{1}{x}}$

5. ## Re: Limit of a function to the power infinite

Originally Posted by Siron
In general, $\displaystyle u=e^{\ln(u)}$, because $\displaystyle y=\ln(u) \Leftrightarrow e^{y}=u \Leftrightarrow e^{\ln(u)}=u$. So for this limit, let $\displaystyle u=\tan\left(\frac{\pi}{4}+x\right)^{\frac{1}{x}}$
Actually that was not my question. I know that $\displaystyle u=e^{\ln(u)}$ but how did you make e the base in this:

$\displaystyle e^{\lim _{x \rightarrow 0}\ln \left( \tan \left( \frac{\pi}{4}+x \right)\right)^\frac{1}{x}$

6. ## Re: Limit of a function to the power infinite

since the exponential function is continuous
lim f(g(x)) = f(lim g(x))
if f is continuous at lim g(x)