Variations of $\lim_{n\rightarrow \infty} \left(1 + \frac{1}{n}\right)^n$

Question

I understand that $$ \lim_{n\rightarrow \infty} \left(1 + \frac{1}{n}\right)^n = e $$

However, how would $ \lim_{n\rightarrow \infty} \left(1 + \frac{1}{3n}\right)^{5n} $ be simplified.

The $5$ can be moved outside the limit:

$$ {\lim_{n\rightarrow \infty} \left[\left(1 + \frac{1}{3n}\right)^{n}\right]^5} $$

But how can you simplify it to the form $e^{\frac{m}{n}}$?

The answer is $ e^{\frac{5}{3}} $, but could someone help me understand the methodology?

Answer 1

Don't move $5$. Move $5/3$ instead: $$\lim_{n\rightarrow \infty} (1 + \frac{1}{3n})^{5n}=\left(\lim_{n\rightarrow \infty} (1 + \frac{1}{3n})^{3n}\right)^{5/3}.$$ You will find the inside limit is $e$.

Answer 2

$\lim_{n\rightarrow \infty} (1 + \frac{1}{3n})^{5n} =\lim_{n\rightarrow \infty} ((1 + \frac{1}{3n})^{3n})^{5/3}= e^{5/3}$

Answer 3

As an alternative

$$\left(1 + \frac{1}{3n}\right)^{5n}=e^{5n\log \left(1 + \frac{1}{3n}\right)}\to e^{5/3}$$

indeed

$$5n\log \left(1 + \frac{1}{3n}\right)=\frac53\cdot\frac{\log \left(1 + \frac{1}{3n}\right)}{\frac1{3n}}\to\frac53 \cdot 1 = \frac53$$

Answer 4

Everybody has given the solution, but this is how it hit me,

$$\lim_{n\to \infty}\biggr(\bigg(1+\frac{1}{3n}\bigg)^n\biggr)^5$$

Let $$3n=t$$

$$\lim_{t\to \infty}\biggr(\bigg(1+\frac{1}{t}\bigg)^{\frac{t}{3}}\biggr)^5$$

$$\lim_{t\to \infty}\biggr(\bigg(1+\frac{1}{t}\bigg)^t\biggr)^\frac{5}{3}$$

$$e^\frac{5}{3}$$

Answer 5

Consider the most general case of $$y=\left(1 + \frac{1}{an}\right)^{bn}$$ Take logarithms $$\log(y)=bn\log\left(1 + \frac{1}{an}\right)$$ Now, use Taylor series $$\log(y)=bn\left(\frac{1}{a n}-\frac{1}{2 a^2 n^2}+O\left(\frac{1}{n^3}\right) \right)=\frac{b}{a}-\frac{b}{2 a^2 n}+O\left(\frac{1}{n^2}\right)$$ Use again Taylor $$y=e^{\log(y)}=e^{\frac{b}{a}}-\frac{b e^{\frac{b}{a}}}{2 a^2 n}+O\left(\frac{1}{n^2}\right)$$ which shows the limit and also how it is approached.