[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]
A limit is the value that a function or sequence “approaches” as the input or index approaches some value.
Limit of function
lim ƒ(x) = L<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
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<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
lim P(N)<br />