Epsilon N Proof for Rational Functions

Make the quantity $\frac{3n + 2}{2n + 5} - \frac{3}{2}$ less than epsilon for $n > N$ in an epsilon-N proof.

In this problem, the main concept being addressed is the epsilon-N definition of limits, which is a fundamental tool in analysis for understanding the behavior of functions as they approach a particular value. Specifically, we are working with a rational function and are tasked with demonstrating that the expression becomes arbitrarily small as the variable n increases beyond some threshold N. This type of problem is essential for understanding other analysis concepts such as continuity and convergence of sequences and functions. In solving problems like this, a systematic method often involves algebraic manipulation to isolate the terms that dominate as n becomes large. Additionally, one needs to bound these terms with respect to epsilon, thereby linking analysis concepts with algebraic skills. Successfully solving problems of this nature strengthens one's ability to transition from intuitive, graphical understandings of limits to formal, rigorous verification. It is beneficial to practice these proofs, focusing on identifying the dominant terms and setting up inequalities appropriately.