Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. (a) \(\lim _{x \rightarrow 2^{+}} f(x)\) (b) \(\lim _{x \rightarrow 2^{-}} f(x)\) where \(f(x)=\left\\{\begin{array}{ll}x & \text { if } x<2 \\ 3 & \text { if } x=2 \\ 4 & \text { if } x>2\end{array}\right.\) (GRAPH CANNOT COPY).

Step-by-Step Solution

Answer

(a) 4; (b) 2.

Step 1: Understand the Problem

We need to determine two one-sided limits of the function \( f(x) \) as \( x \) approaches 2 from the right and from the left. The function \( f(x) \) is piecewise, meaning it has different expressions based on the values of \( x \).

Step 1: Determine \( \lim_{x \to 2^+} f(x) \)

To find the limit as \( x \) approaches 2 from the right, consider the expression for \( f(x) \) when \( x > 2 \). Here, \( f(x) = 4 \) for \( x > 2 \), so as \( x \) approaches 2 from the right, \( f(x) \) approaches 4. Thus, \( \lim_{x \to 2^+} f(x) = 4 \).

Step 2: Determine \( \lim_{x \to 2^-} f(x) \)

For the limit as \( x \) approaches 2 from the left, we use the expression for \( f(x) \) when \( x < 2 \). Here, \( f(x) = x \), so as \( x \to 2 \) from the left, \( f(x) \) approaches 2. Therefore, \( \lim_{x \to 2^-} f(x) = 2 \).

Step 4: Summarize Findings

We found that \( \lim_{x \to 2^+} f(x) = 4 \) and \( \lim_{x \to 2^-} f(x) = 2 \). These one-sided limits indicate that \( \lim_{x \to 2} f(x) \) does not exist because the one-sided limits are not equal.

Key Concepts

Piecewise Functions

Piecewise functions are functions defined by different expressions depending on the value of the input variable. They allow us to model a variety of real-world situations where a single expression isn't sufficient. For the function in the exercise, you can see that:

• If \(x < 2\), then \(f(x) = x\). This means that for any value of \(x\) less than 2, the output of the function is simply \(x\) itself.
• If \(x = 2\), the function jumps to \(f(x) = 3\). Here, the value is explicitly stated.
• For \(x > 2\), the function gives \(f(x) = 4\).

By using piecewise definitions, we create functions that can change behavior at specific points. This is crucial in understanding limits, as the behavior of the function may change drastically right at the point of interest.

One-Sided Limits

One-sided limits are a fundamental concept when working with piecewise functions. Instead of looking at the behavior as \(x\) approaches a point from both sides, we isolate each side:

• A limit from the right, denoted \(\lim_{x \to 2^+} f(x)\), means we only consider the values of \(x\) that are greater than 2, approaching 2.
• A limit from the left, denoted \(\lim_{x \to 2^-} f(x)\), involves values of \(x\) that are less than 2, getting closer to 2.

One-sided limits help in understanding the behavior of functions at points where there might be a discontinuity or a sudden jump in value. Knowing the one-sided limits is key to determining the behavior around critical points and can also reveal if the overall limit exists at a particular point.

Approaching from the Right and Left

When a function behaves differently as you approach a particular point from the right and from the left, it's important to take note of this difference.

Understanding Right-Sided Limits

When \(x\) approaches 2 from the right (i.e., \(x > 2\)), the function in our example gives \(f(x) = 4\). As \(x\) closes in on 2 from this side, you consistently get closer to the value 4.

Understanding Left-Sided Limits

By contrast, when \(x\) approaches 2 from the left (\(x < 2\)), the function follows \(f(x) = x\), meaning that as \(x\) nears 2, \(f(x)\) tends towards 2.Because the value approaching from the right, \(4\), differs from the value from the left, \(2\), the two sides do not meet at the same point. Hence, the overall limit \(\lim_{x \to 2} f(x)\) cannot exist, illustrating the importance of considering each side separately to determine the behavior at discontinuities.