Calculus Limit Debate: Does Lim(x→2) √(4-x²) Exist?

Hey guys! Let's dive into a super interesting calculus conundrum that popped up in a classroom recently. The question revolves around the limit of the function √(4-x²) as x approaches 2. There seems to be a disagreement between a student, who has some experience with Abbott's Analysis, and their calculus teacher. The teacher argues that the limit Does Not Exist (DNE), while the student, drawing from their understanding of real analysis, believes the limit does exist. This is a fantastic opportunity to explore the nuances of limits, domains, and how our understanding of functions can influence our conclusions. So, grab your thinking caps, and let's unravel this mathematical puzzle!

At the core of this debate is the evaluation of the limit: lim(x→2) √(4-x²). To tackle this, we need to carefully consider what a limit truly means and how it applies to this specific function. Remember, the limit of a function as x approaches a certain value (in this case, 2) describes the value that the function approaches as x gets arbitrarily close to that value, but not necessarily the value of the function at that point. This subtle distinction is key to understanding the disagreement.

Let's break down the function: We're dealing with √(4-x²), a square root function. We know that square root functions have a crucial restriction: they can only accept non-negative inputs. This means that (4-x²) must be greater than or equal to 0. If we solve the inequality 4 - x² ≥ 0, we find that -2 ≤ x ≤ 2. This tells us that the domain of the function – the set of all possible input values for x – is the closed interval [-2, 2].

Why is the domain so important? The domain dictates the values of x that we can even consider when evaluating the limit. If we try to plug in values of x outside of this interval, like x = 3, we end up with the square root of a negative number, which is not a real number. This is where the discussion of the subset of R, the set of real numbers, comes into play. We're working within the realm of real numbers, and the function is only defined for x values within its domain.

The teacher's argument that the limit DNE likely stems from a more traditional calculus perspective, focusing on the behavior of the function from both sides of the limit point. In essence, for a limit to exist, the function must approach the same value as x approaches the limit point from both the left (values less than 2) and the right (values greater than 2).

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