Suppose f(x) = {x^2 − 5x + 6 / x − 3 if x≠3
a if x=3.
Determine a value of the constant a for which lim x→3 f(x) = f(3).

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of f(x) as x approaches 3. For the limit to exist, the values of f(x) must approach a specific number as x gets closer to 3 from both sides.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. In this scenario, we need to find a value for the constant 'a' such that the limit of f(x) as x approaches 3 equals f(3), ensuring that the function is continuous at x = 3.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this problem, f(x) is defined differently for x = 3 compared to x ≠ 3. Understanding how to evaluate and manipulate piecewise functions is crucial for determining the appropriate value of 'a' that maintains the function's continuity at x = 3.