Evaluate $\lim_{x\to\infty}(1 + \frac{2}{x})^x$

Question

So I have to evaluate: $$\lim_{x\to\infty}\left(1 + \frac{2}{x}\right)^x$$

This is:

$$\lim_{x\to\infty} e^{x\ln\left(1 + \frac{2}{x}\right)}$$

And:

$$x\ln\left( 1 + \frac{2}{x} \right) = \frac{\ln\left( 1 + \frac{2}{x} \right)}{\frac{1}{x}} $$

So I should apply L'Hospital rule and calculate: $$\lim_{x\to\infty}\frac{\ln\left( 1 + \frac{2}{x} \right)}{\frac{1}{x}}$$

However I'm stuck at this step and I don't know how to move further and land at the final result ( that is, $\displaystyle{\lim_{x\to\infty}}\left(1 + \frac{2}{x}\right)^x = e^2$ )

( Alternative options are also welcome as long as they're more simple and straightforward to apply than L'Hospital rule ).

You can't apply L'Hopital to find the limit $\frac{1+\frac{2}{x}}{\frac{1}{x}}$ because it is not an indeterminate form. — Thomas Andrews, Sep 11 '17 at 19:57
In general, we have $\lim_{x\to\infty}(1+k/x)^x=e^k$ where $k\in\Bbb R$. This follows easily with a few substitutions if you start with the limit definition of $e$. — Prasun Biswas, Sep 11 '17 at 19:59
I guess I have now fixed all issues with this question. I typed it in quite fast but now it's fixed — Ruan, Sep 11 '17 at 20:04
You've changed $1+\frac{2}{x}$ to $\ln(1+\frac{2}{x})$ but you haven't changed the rest of your argument, which would necessarily change. — Thomas Andrews, Sep 11 '17 at 20:07
The key trick is to set $y=\frac{1}x$ then compute the limit $\lim_{y\to 0^+} \frac{\ln(1+2y)}{y}$, which is much easier using L'Hopital. — Thomas Andrews, Sep 11 '17 at 20:11
@ThomasAndrews This is what I was looking for. Write it as an answer and it will become the accepted one — Ruan, Sep 11 '17 at 20:12
Question about the same limit: How to calculate $\lim_{x \to \infty} \left ( \frac{x+2}{x} \right )^{x}=e^{2}$ — Martin Sleziak, Feb 25 '18 at 17:23

score 2 · Answer 1 · answered Sep 11 '17 at 19:58

2

Since $\lim_{x\to\infty}(1 + \frac{1}{x})^x =e$, $\lim_{x\to\infty}(1 + \frac{2}{x})^x =\lim_{x\to\infty}((1 + \frac{2}{x})^{x/2})^2 =e^2 $.

answered Sep 11 '17 at 19:58

marty cohen

107,799

Evaluate $\lim_{x\to\infty}(1 + \frac{2}{x})^x$

1 Answers1