91Ó°ÊÓ

When we discuss the left-hand limit of a function, we're focusing on how the function behaves as the input value, denoted as \(x\), approaches a specific point from the left side. This is particularly useful when studying points of potential discontinuity or rapid change.
For the given example in the exercise, consider \( \lim _{x \rightarrow 2^{-}} f(x)=-3 \). This notation tells us that as \(x\) gets closer to 2 from values less than 2, the value of the function \(f(x)\) approaches -3.

• "Approaching from the left" means selecting values such as 1.9, 1.99, 1.999, etc., which are incrementally closer to 2 but still less than 2.
• Here, the function's curve should dip down towards a y-value of -3 as it gets near \(x=2\) from the left.

This insight allows us to understand a key aspect of the graph's behavior and prepare for any potential discontinuity or jump just before reaching \(x=2\). Understanding this helps ensure that the graph we sketch aligns perfectly with the conditions provided.

The concept of the right-hand limit mirrors that of the left-hand limit, but here we focus on approaching a point from values greater than the point of interest. This helps us understand how the function behaves as \(x\) approaches a certain value from the right side.
For instance, with \( \lim _{x \rightarrow 2^{+}} f(x)=5 \), we're instructed that as \(x\) approaches 2 from numbers like 2.1, 2.01, 2.001, and so on, the function \(f(x)\) heads towards 5.

• "Approaching from the right" involves selecting values greater than 2 but getting closer to 2, indicating the tendency or behavior of the graph's right approach.
• Therefore, on this side of the curve, the graph must shift towards a y-value of 5 as it nears \(x=2\).

Recognizing these tendencies is fundamental when sketching graphs because it contributes to accurately depicting how the graph changes and ensures it meets the condition specified for \(x=2\), completing the picture of how the function behaves on both sides around a specific point.

Discontinuity in a graph tells us about sudden jumps, breaks, or gaps in the function's behavior. It plays a critical role when graphing functions with distinct left and right-hand limits at a point.
In our exercise scenario, the term discontinuity arises around \(x=2\) where the left-hand limit yields a value of -3, and the right-hand limit gives 5.

• This results in a "jump" discontinuity, where between the left and right approaches, the function breaks instead of remaining smooth or connected.
• At \(x=2\), instead of transitioning smoothly from one part to another, the graph rapidly jumps from one y-value to another.

A clear distinction within the graph can thus be visually discerned at this point, indicating that the curve splits, affirming the different values the function aims up to from each direction near \(x=2\). Properly understanding such discontinuities allows for more precise sketching of graphs and accurately represents the function's real behavior at points where the function does not seamlessly align.