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:

lim  ƒ(x) = L
x->c

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

     v(t0):= lim    (t0)−s(t0−Δt)/Δt
            Δt→0s

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 (1/2) if I toss the coin once.

Now if I toss N times P(N) = (1/2)^N
Now this probability would be zero if N is ∞.

Mathematical representation of the question is

P(N) = (1/2)^N find N for which P(N) = 0

and Answer to the given question is

lim  P(N)
N->∞

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