Why are limits considered as a central idea of calculus beyond just mechanical computations? I had my first encounter with Calculus a decade ago. Back then it was purely mechanical. Formulas and rules of derivation and integration were being written on the board without deriving it and were told to compute a bunch of derivatives and integrals without even having a notion of functions, limits and other essential concepts.
Today i have a burning desire to relearn all the forgotton math, calculus in particular.
I'm having a bit of concern with the notion of Limits and Continuity.
Is it really essential that i have solid understanding of limits and continuity? I could teach myself differential calculus after analytically computing limits as well as L'Hopital's rules and up to partial derivatives and basic integrals up to trig substitution. Frankly speaking I still don't have a solid understanding of Limits.
Please explain what a limit is in layman's terms, preferbly with an easy to understand analogy. I looked in wikipedia and searched for a lot of youtube videos, but couldn't make sense. Also explain Continuity in simple terms.
I would also like to know a little bit about Linear and quadratic approximations.
Many people agree that it's indeed possible to keep differentiating and integrating functions without even knowing what a limit is! Big question is why is it considered as a central idea of calculus?
 A: Here is a (possibly over simplified) way to look at continuity and differentiability for functions from $\mathbb{R}$ to $\mathbb{R}$.  
Continuity means that there are no "breaks" or "jumps" in the graph of the function.   If I were to draw the graph of the function, I would be able to do it without lifting my pen.  Continuity also means that around every point, I can choose an interval small enough so that the function varies as little as I want.
A limit will be when we look at the behaviour around a point, but not necessarily at the point itself.  It is very important to note that for a continuous function, $\lim_{x\rightarrow a} f(x)=f(a)$ for every point.  (This can actually be an alternate definition for continuity)
Differentiability means that at every point, if we zoom in really really close, the function looks like a straight line segment.  It means that the function will be continuous, and that there will be no sharp points in the graph of the function.
Hope that helps,
Remark:  This answer may be a bit murky, and I personally suggest learning the $\epsilon$-$\delta$ to fully understand things for yourself.
A: Consider a curve/line/function $L$, not necessarily straight nor continuous. Take a point $a$ that lies in the line $L$. 
Consider two other points $a^-$ and $a^+$, both also lies in $L$. The point $a^-$ lies to the left of $a$ (i.e. $a^- < a$) while the point $a^+$ lies to the right of $a$ (i.e. $a < a^+$).
Take another two points $b^-$ and $b^+$, both also lies on $L$. However, now $b^-$ lies between $a^-$ and $a$ (i.e. $a^- < b^- < a$), while $b^+$ lies between $a$ and $a^+$ (i.e. $a < b^+ < a^+$).
Take another two points $c^-$ and $c^+$, both also lies on $L$. However, now $c^-$ lies between $b^-$ and $a$ (i.e. $b^- < c^- < a$), while $c^+$ lies between $a$ and $b^+$ (i.e. $a < c^+ < b^+$).
If we can repeat this process indefinitely, and at each time $L(x^-)$ and $L(x^+)$ are getting closer and closer to the same point (i.e. as $x^- \to a$ then $L(x^-) \to L(a)$ and as $x^+ \to a$ then $L(x^+) \to L(a)$) as we take more and more steps, then we say that "at point $a$, the line $L$ has the limit $L(a)$", i.e.
$$\lim_{x \to a} \space L(x) = L(a)$$
Next, there are a similar notion of one-sided limits. If you only do this process of taking a point closer and closer from one side, then you have a one-sided limit. There is also the notion of limits of infinities, of what happens to the value of the function as they tend to infinity. 
In summary, limit is the study of the function's behavior as they try to reach a certain point.
A: Remark: If you are not familiar with the notion of the limit of a sequence this answer does not repply to your question.

I assume you are familiar with the notion of the limit of a sequence $(x_{n})
$. The limit of a function $f(x)$ when $x$ tends to $a$ is $b$ if and only
if for any sequence $(x_{n})$ of values of $x$ approaching $a$ but different
from $a$ the corresponding sequence $y_{n}=f(x_{n})$ of values of the
function approaches $b$.

[Edited] A function $f(x)$ is continuous at $a$ if and only if (see figure)


*

*$a$ is on its domain and 

*the difference $k=f(x)-f(a)$ tends to $0$ when the difference $h=x-a$ approaches $0$. (In other words, when for any sequence $h_1=x_1-a,h_2=a-x_2$, $h_3=a-x_3,\dots$ approaching $0$ but different from $0$, the corresponding sequence $k_1=f(x_1)-f(a),k_2=f(x_2)-f(a),k_3=f(x_3)-f(a),\dots$ tends to $0$).


Added: concerning your edit

Many people agree that it's indeed possible to keep differentiating
  and integrating functions without even knowing what a limit is!Big
  question is why is it considered as a central idea of calculus?

It is a central idea for many reasons. The first one is that you need to justify the rules you apply when differentiating a
function. The integration is the reverse process.
A: You may like to go through this Khan academy video to understand limits in very simple terms.In general Khan academy videos on calculus is a great resource for an autodidact.
However,if you want to learn the subject properly with good guidance, I like to suggest you to  the $18.01$SC OCW scholar course on single variable calculus.
