Also, dhiab, I see that you're a Super Member. Maybe this is at least partially a challenge. Can you address that?
Let $\displaystyle \ \ ln(X) \ $ mean $\displaystyle \ \ log_e(X)$.
Let y = $\displaystyle \displaystyle\lim_{n \to \infty}(5n)^{1/n}$
ln(y) = ln[$\displaystyle \displaystyle\lim_{n \to \infty}(5n)^{1/n}]$
ln(y) = $\displaystyle \displaystyle\lim_{n \to \infty}ln[(5n)^{1/n}]$
ln(y) = $\displaystyle \displaystyle\lim_{n \to \infty}\bigg(\dfrac{1}{n}\bigg)ln(5n)$
ln(y) = $\displaystyle \displaystyle\lim_{n \to \infty}\dfrac{ln(5n)}{n}$
I'll use L'Hopital's Rule:
ln(y) = $\displaystyle \displaystyle\lim_{n \to \infty}\dfrac{\bigg(\dfrac{1}{n}\bigg)}{1}$
ln(y) = 0
$\displaystyle e^{ln(y)} = e^0$
y = 1
The limit is 1.
TO: greg1313, O.K. I can understand your being insulted by that comment. But I will add that depending on the level of the course, I would not accepted a solution using differentiation, l'Hopital's Rule. In my analysis courses I always went for more general principles.
This was one of my favorite question. SEE my post above: $\displaystyle \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{n} = 1$
Having proved that, it an easy step to show that for $m\ge 0$ then $\displaystyle \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{m} = 1$
Of course $\sqrt[n]{5n}=\sqrt[n]{5}\sqrt[n]{n}$. Now that is a real teaching moment, a general concept. Not a one time off problem.
You do not understand much about notation do you? In $\displaystyle \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{{n^m}}} = {\left( {\mathop {\lim }\limits_{n \to \infty } \sqrt[n]{n}} \right)^m}$ anyone claiming to help at this level surely realizes that $n$ is a variable but $m$ is constant. So given your limitation, please in the further consider if you are up to answering a question.
You do not understand much about not making presumptions, do you!? So, given your limitation in your lack of ability to make a valid discussion, please refrain from making them,
because you cross into making uncivil posts. Your history of it is clear. Stop doing it, hurry up and understand that. Your post was reported. And you haven't been learning from
from your bad attitude notifications.