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:
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..
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