Conceptual Understanding behind a Limit 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.
 A: Limits are about how functions behave close to but NOT at the chosen point. When you speak about $\lim_{x\to 0} f(x)$ the information you get is about what $f(x)$ does for small nonzero values of $x$. The limit doesn't care about what $f(0)$ or even whether $f(0)$ has a defined value.
It is definitely wrong to think of $\lim_{x\to 0}\frac1x=\infty$ as a claim that $\frac10=\infty$. On the contrary $\lim_{x\to 0} f(x)$ has nothing to do with what $f$ does at $0$ -- only what it does near $0$.
For example, consider $g(x)=\begin{cases} 2 & \text{when }x=0 \\ 1+x & \text{otherwise}. \end{cases}$
(Sketch a graph of this!). Then we have that $\lim_{x\to 0} g(x) = 1$, no matter that $g(0)$ is $2$ rather than $1$.
In the particular case that the function is continuous, the limit $\lim_{x\to a} f(x)$ will be the same as $f(a)$, but that is a special case that you cannot rely on as an explanation of what a limit is -- on the contrary the point that this identity isn't always true, which it why it can be used as a technical definition of "continuous".
But a limit MUST NOT be thought of as just a roundabout way to evaluate the function.

How about limits being infinity? That's because a function can behave in several different ways near-but-not-at $0$. Some of them are common enough that we give them names. Coming near a particular number is one named behavior; we then say that this number is the limit.
But if there's no number the function comes near -- such that in the $\frac1x$ case -- the function may still behave in a way that it is useful to recognize to have a way to speak about. For example $\frac1x$ has the property that when $x$ approaches $0$, the value of $\frac1x$ moves away from every number. This property is what we notate as $\lim_{x\to 0}f(x)=\infty$. It doesn't mean that $\infty$ is a number that $f(x)$ goes towards; $\infty$ is not a number! It's just suggestive notation that is chosen to look like that of $\lim_{x\to 0} g(x)=1$ to make it easier to remember, even though its technical definition is different.
Beware that $\lim_{x\to 0} f(x)=\infty$ does not merely mean that the limit isn't a number. There are functions where $\lim_{x\to 0}$ is neither a number nor $\infty$. For example, consider:
$$ h(x) = \frac{\sin(1/x)}x \quad \text{ defined on } \mathbb R\setminus\{0\} $$
It has the strange property that for every real number we can find a small nonzero $x$ -- which can be as small as we want! -- such that $h(x)$ is that number. This means that $h$ doesn't have any number as its limit for $x\to 0$ -- the informal definition you have seen may not make this clear, in which case you should pester your teacher to explain why not -- and it doesn't have $\infty$ as a limit either.
We would say that $\lim_{x\to 0} \frac{\sin(1/x)}x$ is not defined, which means neither more nor less than we haven't chosen a word or notation for this kind of behavior yet.
If we want to, we can perfectly well choose a notation for speaking about $h$-like behavior -- for example, we could decide to write
$$\lim_{x\to 0} \frac{\sin(1/x)}x ={\updownarrow}$$
Of course we'd need to give a proper definition of what exactly we mean by that, but once that is done $\lim_{x\to 0} \frac{\sin(1/x)}x$ would not be undefined any more.
A: Your question touches one of the basic principles of calculus/mathematical analysis. First I'll try to break down the meaning of a limit (of a sequence) and find a "real world" application for it.
Let's assume you are working in a biology lab with some bacterial culture and you are monitoring the number of living bacteria at any given time $n$. For easier observation purposes let's assume you are checking every hour and count the number of living bacteria, so we have $n=1,2,\dots$. Further we assume that our fairy godmother has provided us with a formula for the number living bacteria at any given time $n$. So if $a_n$ is the number of bacteria after $n$ hours, she told us that $$a_n=\frac{8n^2+n-1}{2n^2-n+5}.$$ We ignore that this would give us some fraction of bacteria and continue working with that. Now we could ask ourselves: will the number of living bacteria increase over time? And if so, will the number grow beyond any limits? While the first question is also important, the second question aims for a (very naive) understanding of the limit of a sequence. Of course this scenario is laughable, but it holds some scientific truth. When you are monitoring something (be it biology, physics, chemistry), you are often not interested at the number of living bacteria at a certain time $n$, but you ask for the long-term development. In our scenario, this long-term development of the number of living bacteria would be the limit of the sequence: $$\lim\limits_{n\to\infty} \frac{8n^2+n-1}{2n^2-n+5}=4.$$
A famous paradox from ancient times when the limit of a sequence wasn't known yet also points out the importance of the concept of a limit.
Starting with the understanding of the limit of a sequence, one can now progress to study (real-valued) functions $f:\mathbb R\rightarrow\mathbb R$. Although one can define the continuity of a function without using limits, in my oppinion the approach using the definition in terms of limits of sequences feels more natural to work with. Same goes for the limit of a function.
Now for your remaining questions:
First I'd like to point out, that your definition of a limit is at least questionable. For the definition of the limit $\lim\limits_{x\to a} f(x)$ there is no need for $f$ to be not defined at $x=a$. The expression "seems to approach" is far away from providing a correct, mathematical view on this subject (e.g. one could say that the function $f:\mathbb R^*\rightarrow \mathbb R,~x\mapsto \frac 1x$ seems to approach $L=-42$ as $x\rightarrow \infty$, as the distance between $\frac 1x$ and $-42$ gets smaller and smaller, but one wouldn't say that this is the limit), but for visualization purposes one can work with that
Second I'd like to say, that (if it happened like you wrote) that your teacher is wrong. The limit of $\frac 1x$ as $x\rightarrow 0$ is not $\infty$ nor is it $-\infty$ or $0$ or anything else, this limit just doesn't exist. Also to use this as a claim to define $\frac 10=\infty$ is not correct.
Now what the limit $\lim\limits_{x\to a} f(x)$ does, is to provide information what your function $f$ does or how it behaves, when you "get near" to $x=a$. Again, this has got nothing to do what $f(a)$ truly is or if it is even defined, all the limit tells you what is happening "close to $x=a$".
Let's use this for your other function, $f(x)=\frac{\sin(x)}{x}$. This function is obviously not defined for $x=0$, but the limit of this function still exists in a "natural way". One can show that if you get closer and closer to $0$, using symbols $x\rightarrow 0$, that $\frac{\sin(x)}{x}$ gets closer and closer to $1$. So in a neighbourhood of $x=0$ (meaning little left or little bit right of $0$ but very close to it), your function will give values very close to $1$ and thus "approach the limit $L=1$".
One closer look at $\lim\limits_{x\to 0} \frac 1x$: it is true, that if you look at positive numbers close to $x=0$, your function will give you very big numbers e.g. for $x=\frac{1}{10}$ you will get $f(x)=10$, for $x=\frac{1}{100}$ you will get $f(x)=100$ etc., so getting "closer" to $0$, your numbers will increase. One might think that $\lim\limits_{x\to 0} \frac 1x=\infty$ meaning that it will grow beyond any limits and leaving every other behind. But this only holds if you use positive numbers. If you take $x=-\frac{1}{10}$ you will get $f(x)=-10$, so your numbers will decrease if you get "closer" to $0$ from the negative direction. So why wouldn't you say $\lim\limits_{x\to 0} \frac 1x = -\infty$? This obviously is a contradiction, thus one says this limit does not exist.
Still it is possible to define $L=\infty$ as the limit of a function, but in general one wants to have the limit be not depending on the side of which you're approaching $a$.
A: 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.
