Find Continuity Constant for Piecewise Function

Find the value of the constant $a$ that will make the piecewise function $f(x) = \begin{cases} ax - 2, & \text{if } x < 3 \\ x^2 - 5, & \text{if } x \geq 3 \end{cases}$ continuous at $x = 3$.

In this problem, you are tasked with ensuring the continuity of a piecewise function at a particular point. Specifically, the goal is to find the value of a constant that makes the piecewise function continuous at the border where the definition of the function changes. This requires an understanding of the definition of continuity at a point, particularly in terms of limits.

A function is continuous at a point if the left-hand limit, right-hand limit, and the function value at that point are all equal. For this piecewise function, setting the pieces equal to each other at the transition point allows you to solve for the unknown constant. This concept is fundamental in real analysis where evaluating limits and ensuring continuity are critical skills.

Think about how the pieces of the function behave as they approach the transition point from either side. Evaluating limits from both the left and the right is key to solving this problem, as continuity is about the smooth transition of function values without any disruptions or jumps. This problem serves as a useful exercise in applying the basic principles of limits and continuity of functions, offering practical insight into analyzing and solving piecewise function continuity issues encountered frequently in real-world applications.