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 ).

Answer 1

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 $.