TITLE: Conceptual Understanding behind a Limit
QUESTION [9 upvotes]: Before I ask anything, let me first express my apologies for this question. 
$$$$When starting with Calculus, my Maths teacher had never felt the need to explain the conceptual understanding behind Limits, or when we take them and why we take them. All that we needed to know (according to him) was:$$$$ "if a function $f(x)$ is not defined at $x=a$ and $f(x)$ seems to approach a certain value L as $x$ approaches $a$, then $L$ is the Limit of $f(x)$ $$$$ Immediately after taking this down, we had been asked to evaluate $\dfrac{1}{x}$ at $x=0$, and the answer we got (contrary to the standard 'Not Defined' answer we had learnt till now) was $\infty$. Without pausing to explain, he had then given us another problem: evaluate $\dfrac{\sin(x)}{x}$ at $x=0$. $$$$
Would somebody please explain what Limits truly are? Why do we need to take Limits? Wen would we use Limits? How can we actually evaluate a function at a point where it does not exist (for example: how is $\dfrac{1}{0}=\infty$?)? $$$$
I woul be truly grateful if somebody could please help clear these doubts I face. Many many thanks in advance.

REPLY [2 votes]: Here, traveler, bolted to the castle wall: the enchanted Crossbow of Augustine. Let me show you how it works.
Notice that the crossbow is always aimed at the horizon. Turning this wheel here turns the bow back and forth—try it. Yes, the relationship between the position of the wheel and the direction of the bow is complicated. That's the price of magic, I guess. The horizon is marked off in degrees, from $-90$ to $90$, and the wheel's got numbers scratched on it too, from $0$ to $1$. Don't know who came up with that, but now we're stuck with it. A wizard once told me I should think of the positions of the wheel as the interval $X = (0, 1)$ and the points on the horizon as the interval $Y = (-90, 90)$, so the relationship between the position of the wheel and the direction of the bow is a function $f$ from $X$ to $Y$. I never understood that, though maybe you do.
Now, here's the tricky bit. If you put the wheel at exactly $\frac{1}{3}$, you can shoot an arrow at $0$ degrees. I've seen it—my buddy Robin can do it, pretty sure she's got some elf blood in her. But normal folks can't put the wheel at exactly anywhere, our hands shake too much. Look, I'll try it. The wheel's at $\frac{1}{3}$, or at least it looks like $\frac{1}{3}$ to me, but—zing—the arrow goes off at around $40$ degrees. That sounds like bad news. The $\frac{1}{3}$ position puts an arrow at $0$ degrees—$f(\frac{1}{3}) = 0$, as the wizard would say—but if you set the wheel even the tiniest bit off, the arrow goes somewhere else.
But there's good news! It turns out that if you set the wheel close enough to $\frac{1}{3}$, the arrow always goes off at around $40$ degrees—unless you put the wheel at exactly $\frac{1}{3}$, of course, but nobody's unlucky enough to do that by accident. Even better: the closer the wheel is to $\frac{1}{3}$, the closer the arrow goes to $40$ degrees. So, with practice, you can set the wheel near enough $\frac{1}{3}$ to hit any creature that wanders across the $40$-degree mark, no matter how skinny it is. The wizard got real excited about that when I showed him, started slapping his knees and laughing, "Oh, of course that trickster would have! Yes,
$$\lim_{x \to \frac{1}{3}} f(x) = 40,$$
that's brilliant!" I don't see why it's funny, but I'm grateful for it: it's what makes the crossbow usable, for ordinary folks like me.
Now, there's some places where the news is not so good. As you turn the wheel toward $\frac{3}{4}$, for example, the bow waggles back and forth between $-15$ and $-10$ degrees, and only sheer luck or magic can aim it any better than that. There's no point you can home in on by getting the wheel close enough to $\frac{3}{4}$. The wizard's eyes gleamed when he saw that. "Ah, looks like $f$ doesn't have a limit at $\frac{3}{4}$. A place where a function has no limit—that's where real magic can happen." I never understood that either, but I was reminded of it years later, during a terrible battle. A whole column of goblins was marching up that pass between $-15$ and $-10$ degrees, and poor Robin with her arm all mangled could barely turn the wheel, but she said "Just put it as close as you can to $\frac{3}{4}$, and I'll do the rest." So we did, and she mowed them all down without hardly touching the wheel. Yeah, sometimes I think this bow wasn't made for clumsy hands like ours.
Well, let's not dwell on that. I'll put the wheel somewhere more sensible. How about $\frac{1}{2}$, that's a fun one! Here, try setting the wheel at almost $\frac{1}{2}$, but just a hair off to the left. You can see the bow's at nearly $-90$ degrees. Now, tap the wheel over so it's still at almost $\frac{1}{2}$, but just a hair off to the right. Now the bow's at nearly $90$ degrees! As the wheel closes in on $\frac{1}{2}$, the bow closes in on $-90$ degrees if you come from the left, and $90$ degrees if you come from the right. With practice, you can hit any creature that steps across the $-90$-degree mark or the $90$-degree mark by setting the wheel close enough to $\frac{1}{2}$; you just have to be careful about which side of $\frac{1}{2}$ you're on. The wizard muttered something complicated about that, like, "Technically $f$ has no limit at $\frac{1}{2}$ either, but it's got the left and right limits
$$
\lim_{x \to \frac{1}{2}{}^-} f(x) = -90
\qquad\text{and}\qquad
\lim_{x \to \frac{1}{2}{}^+} f(x) = 90,
$$
so it's hardly any trouble for you."
It'll be dark soon; you'd better head inside. If you go to the library, you'll find some stuff the wizard left—pictures and scraps of writing he said our crossbow reminded him of, though it's hard to see how.
$\frac{1}{3}$ x | $\frac{3}{4}$ x x | $\frac{1}{2}$ x x x x
No, it's been lovely, but I'm staying out here. Someone's got to turn that wheel.