A limitless dialogue

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Two young mathematicians consider a way to compute limits using derivatives.

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: $\begin{align*} \lim_{x\to 0} \frac{\sin(x)}{x} &= \lim_{x\to 0} \frac{\dd{x}\sin(x)}{\dd{x}x}\\ &= \lim_{x\to 0} \frac{\cos(x)}{1}\\ &=1. \end{align*}$
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 $\begin{align*} \lim_{x\to 0} \frac{\dd{x}\left(x^2+1\right)}{\dd{x}\left(x+1\right)} &= \lim_{x\to 0} \frac{2x}{1} \\ &=0. \end{align*}$

Find five examples where this “trick” works, and five examples where it doesn’t work.

Start with limits of fractions that you know how to compute. Then take the derivative of the numerator and the denominator, and see if the new limit equals the old limit or not.

What is the pattern for when the “trick” works and when it does not work?

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
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factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
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Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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