In my earlier videos I went over limits and limit laws but through a intuitive definition that used subjective terms such as "close to a" but that is not good enough to prove conclusively some limits especially limits that don't have the direct substitution property. In this video I go over the precise definition of a limit as well as going through a simple example to better illustrate it. This concept is more abstract and complex but it is necessary to make defining the limit more concrete. This is very important to understand as it is the building block for calculus so make sure to watch this video!
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Precise Definition of a Limit
Recap on Intuitive Definition:
“As x is close to a, f(x) approaches L”
Need precise definition to prove conclusively limits such as:
Let f be a function defined on some open interval that contains the number ‘a’, except possibly at ‘a’ itself.
Then we can say that the limit of f(x) as x approaches ‘a’ is ‘L’, and we write:
If for every number ε > 0 there is a number δ > 0 such that:
How close to 3 does x have to be if f(x) differs from 5 by less than 0.1?
Thus Precise Definition states that the distance between f(x) and L and be made arbitrarily small by taking the distance from x to a sufficiently small (but not 0).
NOTE: Not using subjective terms such as “close to”.