Problem 57 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(55 - 58 ,\) complete parts \(( a ) - ( d )\) for the piecewise- definedfunction. \(\quad (\) a) Draw the graph of \(f\) . (b) At what points \(c\) in the domain of \(f\) does \(\lim _ { x \rightarrow c } f ( x )\) exist? (c) At what points \(c\) does only the left-hand limit exist? (d) At what points \(c\) does only the right-hand limit exist? $$f ( x ) = \left\\{ \begin{array} { l l } { \sqrt { 1 - x ^ { 2 } } , } & { 0 \leq x < 1 } \\ { 1 , } & { 1 \leq x < 2 } \\ { 2 , } & { x = 2 } \end{array} \right.$$

Step-by-Step Solution

Answer

The graph consists of a half-circle on the interval \( [0,1)\), a horizontal line at \(y=1\) on the interval \([1,2)\), and a point at \(x=2\), \(y=2\). The limit exists at every point except potentially \(x=1\) and \(x=2\). The left-hand limit exists and equals 0 at \(x=1\), and doesn't exist at \(x=2\). The right-hand limit exists and equals 1 at \(x=1\), and doesn't exist at \(x=2\).

Step 1: Graph the Function

It's necessary to plot three separate pieces of the function on the graph: the half-circle function \(f(x) = \sqrt { 1 - x ^ { 2 } }\) for \( 0 \leq x < 1\), the horizontal line \(f(x) = 1 \) for \( 1 \leq x < 2\), and the point at \(x = 2\) where \(f(x) = 2\). The three pieces should be combined into one graph.

Step 2: Identify Where Limits Exist

The limits exist at every point \(c\) where \(f(x)\) is defined, except potentially where \(f(x)\) jumps from one piece to another (at \(x=1\) and \(x=2\)). The limit at these points need to be evaluated by taking the right-hand and left-hand limit.

Step 3: Evaluate the Left-Hand Limit

A left-hand limit only exists at \(x = 1\) and \(x = 2\), as these are the points where \(f(x)\) jumps from one piece to another. For \(x = 1\), the left-hand limit is the limit of \(f(x) = \sqrt { 1 - x ^ { 2 } }\) as \(x\) approaches 1 from the left, which is \( \sqrt{1-1} = 0\). For x = 2, there's no left-hand limit since there's no piece of the function defined for x > 2.

Step 4: Evaluate the Right-Hand Limit

A right-hand limit only exists at \(x = 1\). This is determined by the limit of \(f(x) = 1 \) as \(x\) approaches 1 from the right, which is 1. At \(x = 2\), there's no right-hand limit since that's the extreme right point where the function is defined.

Key Concepts

Limits of Functions

Understanding the limits of piecewise functions is key to analyzing how these functions behave around specific points. A limit describes the value that a function approaches as the input approaches a certain point. Think of it as a way of understanding what happens near to, but not necessarily at, a point.

• Left-Hand Limit: Approaches from the left side. For the function given, we look at values of \(x\) approaching 1 from the left for the half-circle portion \(f(x) = \sqrt{1-x^2}\), which tends toward 0 as \(x\) nears 1, if not reached due to its open endpoint.
• Right-Hand Limit: Approaches from the right side. For \(x = 1\), the step-function piece \(f(x) = 1\) is consistent, so from the right, the function approaches 1 without deviation.

When both the left-hand and right-hand limits agree, the overall limit exists at that point. If they do not agree, there is a discontinuity at that point, often observed where the function has a jump like at \(x=1\) and \(x=2\) in this exercise.

Graphing Functions

Graphing piecewise functions requires attention to how each piece of the function behaves over its specific interval. Begin by plotting each separate expression based on its defined domain:

• Half-Circle Segment: This is defined as \(f(x) = \sqrt{1-x^2}\) and corresponds to the top half of a unit circle over \(0 \leq x < 1\). It visually represents a semicircle that has an open circle at \(x = 1\) indicating not included in the domain of this piece.
• Constant Segment: Transition to \(f(x) = 1\) for \(1 \leq x < 2\). This is a horizontal line indicating constant value over the interval, where an open circle follows the graph at \(x = 2\) since that point isn’t included in that segment.
• Point Segment: At \(x = 2\), \(f(x) = 2\) is simply plotted as a single point, forming a distinct location on the graph to emphasize the jump in value.

By combining these, the overall graph demonstrates the behavior and changes of each function part clearly.

Continuity

Continuity of a function at a point means the function is smoothly connected without breaks, jumps, or holes at that point. For piecewise functions like this, evaluating continuity involves verifying certain conditions related to limits.

• Continuity Requirements: For a function to be continuous at \(c\), \(f(c)\) must be defined, limits from both sides of \(c\) must exist, and those limits must be equal to \(f(c)\).
• Analysis of Given Function: In this example, the function is not continuous at \(x = 1\) or \(x = 2\). At \(x = 1\), the left-hand and right-hand limits of 0 and 1 show a jump, creating a discontinuity. At \(x = 2\), the only defined value is a solitary point with no leading approach or continuation.

Understanding continuity helps in knowing where a function tends to be predictable or where surprising deviations might occur. This particular piecewise function showcases how discontinuities can manifest and how they affect function interpretation.