Saturday, February 10, 2018

Formal definition of limits

The formal definition of limits is mostly skipped in teaching about limits. The intuitive notion is the most taught. The formal definition of limits is interesting and is derived from the intuitive notion.

The intuitive definition says that the limit of a function f is L when x approaches a real number "a" if f becomes closer and closer to "L". We write lim f(x) = L when x➡ a  .

 When x comes closer and closer to a number "a" both to the left and right of that number, x gets values in the neighborhood of a. The variable x in approaching to "a" comes to a certain distance to "a" both from the left and the right. As this distance is very small we call it δ . The variable "x' takes values in the interval ⦐a-ẟ  a+ẟ[. The set of values "x" are translated into the equation ᥣx-aᥣ<ε.

The independent variable f(x) comes to a certain distance of "L" both from the right and the left. As this distance is very small we call it δ. The independent variable f(x) takes values in the interval ]f(x)-ε  f(x)+ε[.

When x approaches "a" f(x) approaches  "L". In other words you give me an ϵ such that ❙f(x)-a❙<ε I will find you a δ that satisfies the equation ❙x-a❙<δ.


The formal definition of limits is stated:


Method

You give me the ε from the equation ❘f(x)-L❘<ε. I'll transform this equation in ❘x-xindice0❘<δ in order to find δ.
















However when the inequality ❘f(x)-L❘<ϵ is transformed in a second degree inequality the process becomes more complicated. Follow the process of solving the example below to solve similar examples..
























Interested in taking online Calculus and tutoring in the same subject visit Center for Integral Development