Part 5/8:
The definition of limits for continuous functions shares a fundamental resemblance to the definition for sequences. If ( F ) is a function defined on an interval ( [a, \infty) ), we write:
[
\text{limit as } x \to \infty \text{ of } F(x) = L
]
This statement is true if for any ( \epsilon > 0 ), there exists a corresponding ( n ) such that if ( x > n ), then ( |F(x) - L| < \epsilon ). Effectively, the sequence defined by ( F(n) = a_n ) translates this definition into the integer domain.