11. Limits(Rigorous Approach)

Formal Definition

Dated: 30-10-2024

For any number \(\epsilon > 0\) , if we can find an interval \((x_0, x_1)\) containing a point \(a\) such that such that

\[L - \epsilon < f(x) < L + \epsilon\]

for all \(x \in (x_0, x_1)\) except possibly \(x = a\) then,

\[\lim_{x \rightarrow a} f(x) = L\]

So, \(f(x)\) exists in the interval \((L - \epsilon, L + \epsilon)\)
The \(\epsilon\) is a very small number. Here is an example

\[\frac{1}{10^{10^{100}}}\]

This below part of fraction is called Googolplex.