Part 6/8:
One significant theorem draws a comparison between functions and sequences. Theorem 1 states that if the limit of a function as ( x \to \infty ) equals ( L ), and if ( a_n ) is defined as ( F(n) ), then the limit of the sequence as ( n \to \infty ) will also equal ( L ).
Through these interpretations, understanding limits simplifies to recognizing the behavior of sequences and functions as they converge to specific values.
Practical Example: Rational Powers
To exemplify this definition, consider the limit:
[
\text{limit as } x \to \infty \text{ of } \frac{1}{x^r} \quad (r > 0)
]