Why isn't $\lim \limits_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}$ equal to $1$? Given $\lim \limits_{x\to\infty}(1+\frac{1}{x})^{x}$, why can't you reduce it to $\lim \limits_{x\to\infty}(1+0)^{x}$, making the result "$1$"?  Obviously, it's wrong, as the true value is $e$.  Is it because the $\frac{1}{x}$ is still something even though it's really small?  Then why is $$\lim_{x\to\infty}\left(\frac{1}{x}\right) = 0\text{?}$$
What is the proper way of calculating the limit in this case?

In the expression
$$\left(1+\frac{1}{x}\right)^x,$$
the $1+1/x$ is always bigger than one.  Furthermore, the exponent is going to $\infty$ and (I suppose) that any number larger than one raised to infinity should be infinity.  Thus, you could just as easily ask, why isn't the limit infinity?
Of course, the reality is that the $1/x$ going to $0$ in the base pushes the limit down towards $1$ (as you observe) and that the $x$ going to $\infty$ in the exponent pulls the limit up to $\infty$.  There's a balance between the two and the limit lands in the middle somewhere, namely at the number $e$, a fact explained many times in this forum.
