Problem 12 - Calculus Early Transcendentals

Question

Sketch the graph of the following function and use it to determine the values of a for which \(\lim _{x \rightarrow a}\) \(f(x)\) exists: $$f(x)=\left\\{\begin{array}{ll}{2-x} & {\text { if } x<-1} \\ {x} & {\text { if }-1 \leq x<1} \\ {(x-1)^{2}} & {\text { if } x \geqslant 1}\end{array}\right.$$

Step-by-Step Solution

Answer

The limit exists for all \(a\) except \(a = -1\) and \(a = 1\).

Step 1: Analyze the Function Definition

The function is defined piece-wise with three different expressions depending on the value of \(x\). For \(x < -1\), \(f(x) = 2 - x\). For \(-1 \leq x < 1\), \(f(x) = x\). Lastly, for \(x \geq 1\), \(f(x) = (x-1)^2\). We will analyze each piece separately before sketching the entire graph.

Step 2: Sketch the Graph of Each Piece

1. For \(x < -1\), the graph of \(f(x) = 2 - x\) is a straight line with slope \(-1\) and y-intercept \(2\).2. For \(-1 \leq x < 1\), the graph of \(f(x) = x\) is a diagonal line passing through the origin.3. For \(x \geq 1\), the graph of \(f(x) = (x-1)^2\) is a parabola starting at \(x=1\) with vertex at \((1, 0)\) and opening upwards.

Step 3: Identify Discontinuities

Check the connection points between the pieces at \(x = -1\) and \(x = 1\). - At \(x = -1\), \(\lim_{x \to -1^-}f(x) = 3\) and \(\lim_{x \to -1^+}f(x) = -1\). Since the limits from left and right differ, \(f(x)\) is discontinuous here.- At \(x = 1\), \(\lim_{x \to 1^-}f(x) = 1\) and \(\lim_{x \to 1^+}f(x) = 0\). The limits from left and right differ, indicating a discontinuity.

Step 4: Determine Points Where Limits Exist

The limit \(\lim_{x \to a} f(x)\) exists for all \(a\) within any continuous piece of the graph. Thus, \(f(x)\) is continuous, and limits exist at all points on the function except at \(a = -1\) and \(a = 1\) due to the discrepancies identified in Step 3.

Step 5: Conclude the Valid Values of a

Given that the limit exists everywhere except where there is a discontinuity, the limit \(\lim_{x \to a} f(x)\) exists for all \(a\) not equal to \(-1\) or \(1\). Therefore, \(a\) can be any real number except \(-1\) and \(1\).

Key Concepts

Limits in Calculus

Limits play a fundamental role in calculus and help us understand the behavior of functions as they approach specific points. For a given function, if you want to find the limit as \( x \) approaches a value \( a \), you essentially examine what happens to the function's value when \( x \) gets very close to \( a \). You do this from both sides of \( a \). The limit exists if the function's value approaches the same result from both the left \(( x \to a^-)\) and the right \(( x \to a^+)\).
Consider our exercise: We examined limits at the points \( x = -1 \) and \( x = 1 \) for a piecewise function. Since from different sides of these points the limits came out different, they illustrated discontinuities at these locations. This means, at those particular points, no single limit value exists.
Understanding this concept is crucial because it lays the groundwork for further studies in calculus, including continuity, derivatives, and integration. By mastering limits, you gain insight into how functions behave near certain points, even before evaluating them directly.

Continuity and Discontinuity

Continuity in functions refers to a smooth, unbroken path on the graph, where you can draw the function without lifting your pencil. In math terms, for a function to be continuous at a point \( a \), the following must be true:

• The function \( f(x) \) is defined at \( x = a \).
• The limit \( \lim_{x \to a} f(x) \) exists.
• The limit matches the function's value, i.e., \( \lim_{x \to a} f(x) = f(a) \).

Discontinuities occur when any of these conditions aren't met. In the original exercise, we witnessed discontinuities at \( x = -1 \) and \( x = 1 \). At these points, the left-hand and right-hand limits didn't match, leading to no single unified value of the function.
Understanding continuity and discontinuity equips you with the skills to identify and analyze breaks in functions. It enables a deep comprehension of function behavior, crucial for both theoretical and practical applications.

Graphing Functions

Graphing functions, especially piecewise functions, is an excellent way to visualize how a function behaves across different intervals. When you handle piecewise functions, you plot each "piece" based on its corresponding condition. This method lets you break down complex functions into simple parts.
In our exercise, we had three pieces:

• A line \( f(x) = 2 - x \) for \( x < -1 \). This piece gave us a downward sloping line.
• A line \( f(x) = x \) for \( -1 \leq x < 1 \), which is a diagonal line through the origin.
• A parabola \( f(x) = (x - 1)^2 \) for \( x \geq 1 \), adding a curve to our graph, starting from \( x = 1 \).

Graphing these segments separately and then connecting them shows how each affects the overall continuity and discontinuity. It's enlightening because it transforms abstract equations into visuals that convey clearer insights into the function's true nature.
Utilizing graphing tools enhances your ability to analyze and interpret mathematical scenarios, particularly when dealing with or investigating changes at specific values.