*[Warning: These are more like my notes which I make public so that if at all there are some stupid people like me, these would come handy]*

**Mathematical definition:**

A limit is the value that a function or sequence “approaches” as the input or index approaches some value.[1]

Limit of function

Mathematical Representation:

```
<br />
lim ƒ(x) = L<br />
x->c<br />
```

This says that L is the value of function ƒ(x) when x reaches to value c.

There is one more definition which is kind of logical evidence to this is L is the limit of function ƒ(x) with ε as a small positive real number then value of ƒ(x) lies in (L-ε,L+ε) or |ƒ(x) – L| < ε

Reallife examples:

I found this amazing question

**Adding the most upvoted example here**

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 of your travel the limit

```
<br />
v(t0):= lim (t0)−s(t0−Δt)/Δt<br />
Δt→0s<br />
```

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

**And the most simple to understand (for me)**

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

Possibility of getting a head is (½) if I toss the coin once.

Now if I toss N times P(N) = (½)^{N}

Now this probability would be zero if N is ∞.

Mathematical representation of the question is

P(N) = (½)^{N} find N for which P(N) = 0

and Answer to the given question is

```
<br />
lim P(N)<br />
N->∞<br />
```