Proving a Limit with EpsilonDelta Definition

Prove that $\lim_{x \to 2} (3x + 1) = 7$ using the epsilon-delta definition of a limit.

This problem involves using the epsilon-delta definition of a limit, which is a fundamental concept in understanding the rigorous approach to limits in calculus and real analysis. The epsilon-delta definition allows us to precisely define what it means for a function to approach a limit as the input approaches a certain point. In solving such problems, it's crucial to understand the relationship between epsilon, which represents how close you want the function value to be to the limit, and delta, which indicates how close the input needs to be to the target point to achieve this.

The process generally involves choosing a specific epsilon greater than zero and then finding a corresponding delta such that for all x within a delta-neighborhood of the target point, the difference between the function value and the limit is less than epsilon. This rigorous method highlights the importance of the logical structure underlying calculus, distinguishing it from mere heuristic understanding.

In the context of this problem, we explore the linear function 3x + 1. Linear functions have a straightforward relationship between x and 3x + 1, which simplifies finding an appropriate delta for a given epsilon. The exercise of proving limits using the epsilon-delta method is foundational in real analysis as it reinforces a deep understanding of continuity and provides a template to approach more complex functions.