calculus - 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.

Answer

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.

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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.