Part 1/6:

Understanding Limits and Convergence in Sequences

The concept of limits is fundamental in calculus and sequences, playing a pivotal role in understanding how functions behave as they approach points or infinity. In this article, we explore several limit problems and sequence behaviors, illustrating key theorems and principles crucial for students and mathematicians alike.

Limit of a Rational Function

Let’s begin with finding the limit of the sequence defined by ( \frac{n}{n + 1} ) as ( n ) approaches infinity. Upon substitution, we initially observe the form ( \frac{\infty}{\infty} ), which is undefined. To resolve this, we can apply the technique of factoring out the highest power in the denominator.

Rewriting this limit gives us:

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