What limit "identities" are good to memorize for early college calculus? I don't know if "identity" is the correct word, but this would be an example: $$\lim_{x \to \infty} \left(1 + \frac{1}{x} \right)^{x} = e $$
With my current knowledge, I wouldn't really know where to begin in solving this, so whenever I see this notation (or something similar like $\lim_{x \to \infty} \left(1 - \frac{1}{x} \right)^{x} = \frac{1}{e} $), I know I'm dealing with some form of $e$ (if that makes sense).
Other properties I have memorized are for example what the limit of $\arctan$ is (even though I could simply visualize it, we can't using graphing calculators on the exam and I'm very bad at sketching more complex functions), or that $\lim_{x \to 0} \frac{a}{x} \sin{x} = a$. I also just learned about the Stirling approximation from another user here.
Are there any other noticeable limits like these that I should be on lookout for? 
 A: Hint
From the other user !!
$$A= \left(1 + \frac{1}{x} \right)^{x}$$ $$\log(A)=x\log\left(1 + \frac{1}{x} \right)$$ Now, remember that when $y$ is small $\log(1+y)\approx y$ Replace $y$ by $\frac 1x$ and you are done.
What you must know, remember and use are Taylor series for the basic functions.
A: Note: this is not an answer. I just didn't want to deal with writing latex in the comments.
A few off the top of my head are $\lim_{x\to 0} {\dfrac{sinx}{x}} = 1$ and $\lim_{x \to 0}{\dfrac{e^x - 1}{x} = 1}$
Also $\lim_{x \to a}{(1+\dfrac{1}{f(x)})^{f(x)} = e}$ and $\lim_{x \to a}({\dfrac{f(x)}{g(x)}})^{h(x)} = e^{\lim_{x \to a}{h(x)(\dfrac{f(x)}{g(x)} - 1)}}$
A: If you will be encountering sequences and series this is an important one to know:
$$\sum_\limits{n=1}^{\infty} \frac{1}{n} =\lim \limits_{n \to \infty} (1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n}) \rightarrow \infty$$
This is the harmonic sequence, and is very important in determining the convergence of infinite series. The limit can be shown easily using the integral test. 
Edit: I also think that this is particularly important:
$$\lim \limits_{n \to \infty} \frac{A}{n}=0$$ Where $A$ is any real number. Intuitively, this means that an finite number "divided by infinity" will be nothing. That is, a finite quantity, no matter how large, divided by an infinite "quantity" is zero.
