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?

A number of people have expressed the error in your reasoning.  Here's an intuitive reason.
Assume you make a loan of $1$ dollar to a person at 100% interest over a year.  At the end of the year, you get $2$ dollars back.
Now, what if that interest was instead "compounded" over $n$ periods?
For example, if $n=12$, then we'd divide the year into 12 periods. The person would pay $100\%/12$ interest each period on the total due. Assuming the person does not pay any of the loan off, at the end of the year the total due is: $$\left(1+\frac{1}{12}\right)^{12}$$
Now keep increasing the number of compounding periods. Intuitively, the more periods, the more the person owns at the end of the year.
Another way to see this is to look at:
$$\left(1+\frac{1}{2n}\right)^2 = 1+\frac{1}{n} + \frac{1}{(2n)^2}>1+\frac{1}n$$
Raising both sides to the $n$th power, we see: $$\left(1+\frac{1}{2n}\right)^{2n}>\left(1+\frac{1}{n}\right)^n$$
