Find the value of a limit using a known limit

Question

I really have two questions:

Given that

$\lim_{n\to \infty}\frac{\ln n}{n}$= 0,

how is it possible to find the value of:

$\lim_{n\to \infty}\frac{\ln (n + k)}{n}$, where $k > 0$?

Similarly, given that

$\lim_{n\to \infty}\sqrt[n]{n} $ = 1,

how could I go about finding the value of:

$\lim_{n\to \infty}\sqrt[n]{n\cdot(\frac{5}{n})^n} $?

Thank you!!!

Answer 1

By a translation of the variable,

$$\lim_{n\to\infty}\frac{\ln(n+k)}n=\lim_{n\to\infty}\frac{\ln(n)}{n-k}=\lim_{n\to\infty}\frac{\ln(n)}n\frac n{n-k}=\lim_{n\to\infty}\frac{\ln(n)}n\lim_{n\to\infty}\frac n{n-k}=\lim_{n\to\infty}\frac{\ln(n)}n.$$

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Then, $$\lim_{n\to\infty}\sqrt[n]{n\left(\frac5n\right)^n}=\lim_{n\to\infty}\sqrt[n]{n}\frac5n=\lim_{n\to\infty}\sqrt[n]{n}\lim_{n\to\infty}\frac5n=0$$ provided the first limit is finite.

Answer 2

For the first:

$$\frac{\ln(n+k)}{n} = \frac{\ln(n+k)}{n+k}\cdot \frac{n+k}{n}$$

For the second, $$\sqrt[n]{n\cdot \left(\frac5n\right)^n} = \sqrt[n]{n}\cdot \frac5n.$$