Akshay Deo

Do(nt) Code Seriously

Limits (Mathamatics)

[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 />