Check out this dialogue between two calculus students (based on a true story):

Devyn

Yo Riley, guess what I did last night?

Riley

What?

Devyn

I was doing some calculus.

Riley

That. Is. Awesome.

Devyn

I know! Anyway, I noticed something kinda funny. I think you can sometimes take limits by taking the derivative of the numerator and the denominator.

Riley

That’s crazy.

Devyn

I know! But check it: \begin{align*} \lim _{x\to 0} \frac {\sin (x)}{x} &= \lim _{x\to 0} \frac {\frac {d}{dx}\sin (x)}{\frac {d}{dx} x}\\ &= \lim _{x\to 0} \frac {\cos (x)}{1}\\ &=1. \end{align*}

Riley

Woah. That. Is. Awes…weird. Hmmm, but it seems like cheating. Wait, it doesn’t always work, check this out:

\[ \lim _{x\to 0} \frac {x^2+1}{x+1} = 1, \]

but \begin{align*} \lim _{x\to 0} \frac {\frac {d}{dx}\left (x^2+1\right )}{\frac {d}{dx}\left (x+1\right )} &= \lim _{x\to 0} \frac {2x}{1} \\ &=0. \end{align*}