Show your work and calculate limit of the following questions.
For the first problem, which do you mean?
$\displaystyle \lim_{x \to \infty} 3 \sqrt{n}^\frac {1} {2n} $
or
$\displaystyle \lim_{x \to \infty} 3 \sqrt{n}^{\frac {1} {2} n} $
2nd problem
$\displaystyle \lim_{x \to \infty} (n+1)^\frac {1} {ln(n+1)} $
Take the ln of the lim, just remember to rise it to e later.
$\displaystyle \lim_{x \to \infty} \frac {ln(n+1)} {ln(n+1)} $
I took the ln for you in problem 2...I guess i'll show it to you step by step
$\displaystyle \lim_{x \to \infty} (n+1)^\frac {1} {ln(n+1)}$
Just remember to rise the answer to e.
$\displaystyle \lim_{x \to \infty} ln(n+1)^\frac {1} {ln(n+1)}$
By the law of lns..
$\displaystyle \lim_{x \to \infty} \frac {1} {ln(n+1)}ln(n+1)$
$\displaystyle
\lim_{x \to \infty} \frac {ln(n+1)} {ln(n+1)}=1
$
So the answer will be $\displaystyle e^1 $ or just plain e.
Try the first problem this way. You will probably need to l'hospital it.
Have you tried it? I just did and you don't actually need l'hospital.
$\displaystyle
\lim_{n \to \infty} 3 \sqrt{n}^\frac {1} {2n}
$
$\displaystyle
3 \lim_{n \to \infty} \sqrt{n}^\frac {1} {2n}
$
$\displaystyle
3 \lim_{n \to \infty} \frac {ln\sqrt {n}} {2n}
$
It should be obvious from here. Remeber just rise the limit to e, not including the 3.
$\displaystyle
3 \lim_{n \to \infty} \frac {\ln {n}} {4n}
$