Approaching Infinity - Finding the Limit of a Sequence

[alexmahone]

Find $\displaystyle\lim_{n\to\infty}\frac{\sqrt[n]{n}}{\sqrt[n+1]{n+1}}$.

 

[Evgeny.Makarov]

$\sqrt[n]{n}=e^{\frac{\ln n}{n}}$ and $\frac{\ln n}{n}\to 0$ as $n\to\infty$.

 

[alexmahone]

$\sqrt[n]{n}=e^{\frac{\ln n}{n}}$ and $\frac{\ln n}{n}\to 0$ as $n\to\infty$.

So are you saying that the answer is $\displaystyle\frac{1}{1}=1$?

 

[Plato2]

Find $\displaystyle\lim_{n\to\infty}\frac{\sqrt[n]{n}}{\sqrt[n+1]{n+1}}$.

Do you know this limit
$\displaystyle\lim _{u \to \infty } \sqrt{u}=~?$

 

[alexmahone]

Do you know this limit
$\displaystyle\lim _{u \to \infty } \sqrt{u}=~?$

It's 1.

 

[Plato2]

It's 1.

What is the answer to the OP?

 

[alexmahone]

What is the answer to the OP?

$\displaystyle\frac{1}{1}=1$

 

[Evgeny.Makarov]

I agree.