91Ó°ÊÓ

When analyzing limits of piecewise functions, it's crucial to consider what happens as the input values approach specific points. The limit of a function \(\lim_{x \to c} f(x)\) exists if both the left-hand and right-hand limits are equal at that point.
For example, in our function, if we approach a point from the left side, we refer to the left-hand limit \(\lim_{x \to c^-} f(x)\), and from the right, the right-hand limit \(\lim_{x \to c^+} f(x)\).
In the given problem:

• From \(x = 0\) to \(x = 1\), the semicircle is continuous, meaning the limit exists for this interval as each value before 1 smoothly leads into the next.
• At \(x = 1\), though, we have a transition from the semicircle to a constant line, causing a disruption in continuity. Here, \(\lim_{x \to 1^-} f(x) = 0\) from the semicircle and \(\lim_{x \to 1^+} f(x) = 1\) from the constant function, hence the overall limit does not exist.
• At \(x = 2\), we encounter another gap since \(\lim_{x \to 2^-} f(x) = 1\) but \(f(2) = 2\), creating a discontinuity.

Understanding these limits is key to mastering function analysis.

The domain and range of a function give us critical insights into where the function exists and its possible output values. For the piecewise function given, the domain is found by identifying all the x-values where the function is defined.
In our case, the domain is \[ [0, 2] \] because the function covers a semicircle, a line segment, and includes a specific point at \(x = 2\). The entire span from 0 to 2 is covered, though certain points like \(x = 1\) and \(x = 2\) have unique definitions.

• For \(0 \leq x < 1\), it lies on a semicircle with \(f(x) = \sqrt{1-x^2}\).
• For \(1 \leq x < 2\), \(f(x) = 1\) over the straight line segment.
• Directly at \(x = 2\), \(f(x) = 2\), creating a single isolated point on the graph.

By evaluating all these, the range spans from the lowest to the highest outputs, documented as \[ [0, 2] \] due to the semicircle contributing \[ [0, 1] \], the constant \[ 1 \], and point \[ 2 \]. Understanding domain and range sets the stage for analyzing more complex aspects like limits.

Graphing functions provides a visual representation of how they behave and allows us to interpret them much more easily. In the piecewise function provided, we translate its definitions into segments on a graph.
Such translation involves sketching:

• A semicircle, rising from \(x = 0, y = 1\) and dropping to \(x = 1, y = 0\). Don't connect \(x = 1\) because it's a separate component.
• A horizontal line at \(y = 1\) from \(x = 1\) to just before \(x = 2\), and finally,
• A singular point at \(x = 2, y = 2\), indicating a distinct value at this point.

This kind of breakdown reveals where the function is continuous and places where it jumps or gaps occur. Carefully placing these pieces helps in accurately understanding the entire behavior and limits of the function.

Continuity refers to a function being uninterrupted as we move along its graph. In a piecewise function, this isn't always the case. We look for spots where the function is not seamless. In our function:

• The segment between \(x = 0\) and \(x < 1\) is continuous because the semicircle does not face any breaks.
• At \(x = 1\), there's a switch from the semicircle to a line, creating a jump and resulting in discontinuity. There's no seamless transition since \(\lim_{x \to 1^-} f(x) eq \lim_{x \to 1^+} f(x)\).
• Similarly, at \(x = 2\), the function jumps from the value on the line \(\lim_{x \to 2^-} = 1\) directly to \(f(2) = 2\), marking another discontinuity.

Analyzing for continuity is essential, as it affects how we apply the function to models, ensuring we find parts where the graph has no breaks, jumps, or holes. Understanding and identifying discontinuities allows for accurate math modeling.