Problem 25 - Precalculus: Mathematical for Calculus

Question

Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist. $$f(x)=\left\\{\begin{array}{ll} -x+3 & \text { if } x<-1 \\ 3 & \text { if } x \geq-1 \end{array}\right.$$ (a) \(\lim _{x \rightarrow-1^{-}} f(x)\) (b) \(\lim _{x \rightarrow-1^{+}} f(x)\) (c) \(\lim _{x \rightarrow-1} f(x)\)

Step-by-Step Solution

Answer

(a) 4, (b) 3, (c) does not exist.

Step 1: Analyze the Piecewise Function

The function is defined in two pieces. For values of \(x < -1\), the function is \(f(x) = -x + 3\). For values of \(x \geq -1\), the function is constant at 3, so \(f(x) = 3\).

Step 1: Graph the Function for \(x < -1\)

For \(x < -1\), graph the line \(f(x) = -x + 3\). This line will be a straight line with a slope of -1 and a y-intercept of 3. Plot this line for \(x < -1\). The graph will be continuous and approach the point (-1, 4).

Step 2: Graph the Function for \(x \geq -1\)

For \(x \geq -1\), the function \(f(x) = 3\) is a horizontal line at y = 3. Start this line at the point (-1, 3), where it includes the point because \(x = -1\) is part of this piece.

Step 3: Determine \(\lim _{x \rightarrow-1^{-}} f(x)\)

Approach \(x = -1\) from the left. The function is \(-x + 3\) for \(x < -1\). As \(x\) approaches -1 from the left, \(-x + 3\) approaches \((-(-1)) + 3 = 4\). Therefore, \(\lim _{x \rightarrow-1^{-}} f(x) = 4\).

Step 4: Determine \(\lim _{x \rightarrow-1^{+}} f(x)\)

Approach \(x = -1\) from the right. The function is \(3\) for \(x \geq -1\). As \(x\) approaches -1 from the right, the function remains constant at 3. Therefore, \(\lim _{x \rightarrow-1^{+}} f(x) = 3\).

Step 5: Determine \(\lim _{x \rightarrow-1} f(x)\)

Determine if the two-sided limit exists by checking if the left-sided and right-sided limits at \(x = -1\) are equal. Since \(\lim _{x \rightarrow-1^{-}} f(x) = 4\) and \(\lim _{x \rightarrow-1^{+}} f(x) = 3\), the two-sided limit does not exist because the limits from the left and right are not equal.

Key Concepts

Graphing Functions

When we talk about graphing functions, we're referring to the process of drawing a visual representation of a function on a coordinate plane. This helps us understand the behavior of the function across different values of x. For a piecewise function like the one given in the exercise, different rules apply in different intervals of the domain. This means, the graph consists of different segments joined at certain points.

• For the interval where \(x < -1\), the graph is a line described by the equation \(f(x) = -x + 3\). This line has a slope of -1 and a y-intercept of 3.
• For the interval where \(x \geq -1\), the function becomes constant, \(f(x) = 3\), forming a horizontal line.

When plotting these segments, look for special points where the intervals change, like \(x = -1\) in this function, to ensure accuracy. The graph assists in visualizing how these two parts relate to each other, and it's essential for further analysis.

Limits

In calculus, the concept of limits helps us understand the behavior of functions as they approach a particular point. For piecewise functions, limits can tell us how the function behaves as it approaches the point where the rule changes. To find limits like \(\lim_{x \to -1^-} f(x)\), we approach the point from the left side. In our example, as x approaches -1 from the left, we observe the line segment \(f(x) = -x + 3\), which approaches the value 4. Similarly, \(\lim_{x \to -1^+} f(x)\) involves approaching from the right side, where the value is consistently 3 as the line is constant.

• Left-sided limit: The value that \(f(x)\) approaches as x gets close to -1 from values less than -1.
• Right-sided limit: The value that \(f(x)\) approaches as x gets close to -1 from values greater than -1.
• A limit exists at a point if both the left-sided and right-sided limits are equal.

Unfortunately, since the values from the left and right are different (4 and 3 respectively), the two-sided limit \(\lim_{x \to -1} f(x)\) does not exist.

Continuity

Continuity in mathematics refers to a function having no breaks, jumps, or holes at a given point within its domain. A function is said to be continuous at a point \(c\) if the line can be drawn at that point without lifting the pencil, which implies a smooth transition across the value. Mathematically, this happens if \(f(c)\) is defined and equals the limit of \(f(x)\) as x approaches c from any direction.

For the piecewise function described:

• \(f(x)\) is not continuous at \(x = -1\) because the limit from the left (4) does not equal the function value (3), nor does it equal the limit from the right (3).
• To achieve continuity at a point, the function's value and the limits must all agree at that point.

Because the limits are different from each other at \(x = -1\), there is a jump on the graph, indicating the lack of continuity at this key transition point. This piecewise function is an example of how discontinuities can appear when switching between functional rules.