One last question on the concept of limits I read through this post on the notion of limits Approaching to zero, but not equal to zero, then why do the points get overlapped?
But there's one last question I have. [Trust me, it'll be my last ever question on limits!]
 It clearly says as you get closer and closer to a certain point you are eventually getting closer to the limiting point. I.e., as $\Delta x$ approches $0$, you get to the limit. 
Let me give you an example which supports the above statement. Let's say you want to evaluate the limit $$ \lim_{x \to 2} \frac{x^2 - 4}{x-2} .$$ Sooner or later, you have to plug in $2$ and you get the answer $4$ and you say that the limit of that function as $x \to 2$ is $4$. But why should I plug in $2$ and not a number close to $2$? $x$ is certainly not equal to $2$, right? 

An important point to consider is that you cannot actually "plug in" $2$ and get $4$.
When $x = 2$, you will have a division by zero, since $x - 2 = 0$.
When calculating $\frac{x^2 - 4}{x - 2}$ for $x = 2$ using a computer, you may get $4$ due to rounding errors when you have set $x$ to a value very close to 2. This is due to the very fact that the function approaches $4$ as $x$ approaches $2$.
