Finding limit of $\left(1 +\frac1x\right)^x$ without using L'Hôpital's rule. Is there a way to find this limit without using L'Hôpital's rule. Just by using some basic limit properties.
$$\lim_{x\to\infty}\left(1+\frac1x\right)^x=e$$

In the comments it has been pointed out that this usually serves as a definition of e.  But I think I know what you're asking.  Usually in introductory calculus classes to evaluate this limit you let $$y= \lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x,$$ and then take the natural log of both sides to get $$\ln(y) = \lim_{x\to\infty} x\ln\left(1+\frac{1}{x}\right)=\lim_{x\to\infty}\frac{\ln\left(1+\frac{1}{x}\right)}{\frac{1}{x}},$$ where the last limit you use L'Hôpital's rule to get that it is equal to 1.  So I think maybe what you are actually asking is can, $$\lim_{x\to\infty} x\ln\left(1+\frac{1}{x}\right)$$ be evaluated without L'Hôpital's rule?  Please correct me if I'm wrong.  If this is indeed your question you can expand $\ln\left(1+\frac{1}{x}\right)$ as a series at $x=\infty$ which I think makes the limit trivial.  
I don't think I've ever used these kinds of power series so I would appreciate any input about this last suggestion.
