The Reality of Mathematical limit

When I was in high school, I had my days just learning all I can in Mathematics, getting all kinds of credit for theoretical problems. But not until now I start wondering if all we learned then do have their applications in the real world, what would be of limits in Calculus?.

Theories tells that it explains the non-exceeding part of a function, a point where the graph of the function can't moved further again maybe cut off by an asymptote or whatever, or a point in infinity and all sort of explanatory theories like that. Until now, I think I figured it out.

This morning, I read Wikipedia's informal definition of a limit:

Informally, a function f assigns an output f(x)$f(x)$ to every input x$x$. The function has a limit L$L$ at an input p$p$ if f(x)$f(x)$ is "close" to L$L$ whenever x$x$ is "close" to p$p$. In other words, f(x)$f(x)$ becomes closer and closer to L$L$ as x$x$ moves closer and closer to p$p$.

To me that sounds like something that might be better described as a 'target'.

If I take a simple function, say one that only multiplies the input by 2$2$; and if my limit is 10$10$ at an input 5$5$: then I've described something that seems to match the elements contained in Wikipedia's definition. I don't believe that that's right. To me it looks like an elementary-algebra problem (2p=10$2p=10$). To make it more calculusy, I could graph the function's output when I use inputs other than p$p$, but that really wouldn't give me anything but an illustration of the fact that one's answer moves farther from the right answer as it becomes more wrong (go figure).

So limits are important; what I've just described is trivial. I do not understand them. I know calculus is often used for solving real-world challenges, and that limits are an important element of calculus, so I assume there must be some simple real-world examples of what it is that limits describe.

What is a function in a real life definition?, it is a generalized relationship between unknown values of object or an instance.

On the other hand, limit is the phenomenon of finding a point or points in time of the unknown where the function won't yield a real or reasonable result in the relationship.

Limits are super-important in that they serve as the basis for the definitions of the 'derivative' and 'integral', the two fundamental structures in Calculus! In that context, limits help us understand what it means to "get arbitrarily close to a point", or "go to infinity". Those ideas are not trivial, and it is hard to place them in a rigorous context without the notion of the limit. So more generally, the limit helps us move from the study of discrete quantity to continuous quantity, and that is of prime importance in Calculus, and applications of Calculus.

To apply this notion to physics (yes, I'm moving away from math now), it is possible to apply a continuous analysis to motion. We'd like to be able to measure instantaneous speed, which requires the notion of an instantaneous value. Now this is dependent on the concept of the limit. That is to say, we want to measure a quantity in an instant, and we define this "instant" by a limit, i.e., as an approach towards some infinitesimal time. This is how we would answer, e.g., the commonplace question "how fast was he going at time x$x$?".

Here is an fair example very easy to grasp:

If I keep tossing a coin as long as it takes, how likely am I to never toss a head?

If I toss a coin N$N$ times, what is the probability p(N)$p(N)$ that I have not yet tossed a head? Now what is the limit as N→∞$N\to \mathrm{\infty }$ of p(N)

$p(N)$? 

because p=12,14,18,116,…$p=\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},\dots$ gets closer and closer to zero as N$N$ gets "closer to ∞

$\mathrm{\infty }$". 

Another one to take is the reading of your speedometer (e.g., 85 km/h) is a limit in the real world. Maybe you think speed is speed, why not 85 km/h. But in fact your speed is changing continuously during time, and the only "solid", i.e., "limitless" data you have is that it took you exactly 2 hours to drive the 150 km from A to B. The figure your speedometer gives you is at each instant 

t0$t_0$ of your travel the limit

where s(t)$s(t)$ denotes the distance travelled up to time t.