Problem 88 - Single Variable Calculus

Question

In Exercises 88 and 89, plot the graph of \(f .\) Then use the graph to determine where the function is continuous. Verify your answer analytically. \(f(x)=\left\\{\begin{array}{ll}\frac{x+1}{x \sqrt{1-x}} & \text { if } x<1 \\\ 2 & \text { if } x=1 \\ \frac{x^{4}+1}{x^{2}} & \text { if } x>1\end{array}\right.\)

Step-by-Step Solution

Answer

The function \(f(x)\) is continuous for \(x<1\) and \(x>1\), with the graphs of \(f(x) = \frac{x+1}{x \sqrt{1-x}}\) and \(f(x) = \frac{x^{4}+1}{x^{2}}\) being continuous in their respective domains. However, the function is non-continuous at \(x=1\) as the left limit is 1, while the right limit and the function value are both 2.

Step 1: Plot the function graphs

Plot the graphs of the functions for their respective domains: 1. \(f(x) = \frac{x+1}{x \sqrt{1-x}}\) if \(x<1\) 2. \(f(x) = 2\) if \(x=1\) 3. \(f(x) = \frac{x^{4}+1}{x^{2}}\) if \(x>1\) Note: The actual plotting of graphs requires graphing tools or software, which can't be shown in text format. Make sure to plot the graphs using a graphing calculator or software such as Desmos or Geogebra.

Step 2: Analyze continuity on the graph

Observe the plotted graphs in step 1 to analyze where the function is continuous. 1. For \(x < 1\), the graph of \(f(x) = \frac{x+1}{x \sqrt{1-x}}\) seems to be continuous. 2. For \(x = 1\), the function is defined, but check if the left and right limits match the function value. 3. For \(x > 1\), the graph of \(f(x) = \frac{x^{4}+1}{x^{2}}\) seems to be continuous.

Step 3: Analyze the limits for continuity

Now let's verify the limits analytically for all three domains: 1. Determine if the function is continuous for \(x < 1\) For \(x < 1\), analyze the limit as \(x\) approaches \(1^{-}\) of the function \(f(x) = \frac{x+1}{x \sqrt{1-x}}\): \(\lim_{x \to 1^{-}} \frac{x+1}{x\sqrt{1-x}}\) As \(x\) approaches \(1^{-}\), we can conclude that the limit exists and is equal to 1. 2. Determine if the function is continuous for \(x=1\) If the limit as \(x\) approaches \(1\) exists and equals the value of the function at \(x=1\), then the function is continuous at \(x=1\). Since we found that the limit as \(x\) approaches \(1^{-}\) is 1, now determine the limit as \(x\) approaches \(1^{+}\) of the function \(f(x) = \frac{x^{4}+1}{x^{2}}\): \(\lim_{x \to 1^{+}} \frac{x^{4}+1}{x^{2}}\) As \(x\) approaches \(1^{+}\), we can conclude that the limit exists and is equal to 2. So, the left limit at \(x=1\) is 1 while the right limit and the function value are 2, which means the function is not continuous at \(x=1\). 3. Determine if the function is continuous for \(x > 1\) For \(x > 1\), analyze the limit as \(x\) approaches \(1^{+}\) of the function \(f(x) = \frac{x^{4}+1}{x^{2}}\). It has already been determined that it exists and equals 2. Therefore, the function is continuous for \(x > 1\).

Step 4: Conclusion

Based on the analytical verification of the plot, we conclude that the given function is continuous for \(x<1\) and \(x>1\), but non-continuous at \(x=1\).

Key Concepts

Graphical Analysis

Understanding the continuity of a piecewise function often begins with a visual approach. By plotting the different segments of the function, we can observe apparent breaks in the graph. For the function given,

• For values of \(x < 1\), we plot the graph of \(f(x) = \frac{x+1}{x \sqrt{1-x}}\).
• At \(x = 1\), the function is a constant, namely \(f(x) = 2\).
• And for \(x > 1\), the graph of \(f(x) = \frac{x^{4}+1}{x^{2}}\) is considered.

By observing these plots using graphing software, we can intuitively identify where the function appears to be continuous and where it potentially has breaks. This approach allows us to see that there are no apparent discontinuities for \(x < 1\) and \(x > 1\), yet there might be an issue at the point \(x = 1\). Hence, graphical analysis is a preliminary step in examining the behavior of piecewise functions.

Limits

In mathematics, limits play a crucial role in rigorously examining continuity. Analyzing limits involves exploring the values that a function approaches as the input approaches a particular point. For our piecewise function:

• As \(x\) approaches \(1^{-}\), we utilize the segment \(f(x) = \frac{x+1}{x \sqrt{1-x}}\), resulting in a limit of 1.
• For \(x\) approaching \(1^{+}\), the segment \(f(x) = \frac{x^4 + 1}{x^2}\) gives a limit of 2.

These computations confirm that the function’s behavior differs from one side of \(x = 1\) to the other. To be continuous at \(x = 1\), the function must have matching left- and right-hand limits that also equal the function's value at that point. Here, since the left limit is 1, the right limit is 2, and the function's value is 2, the function is not continuous at \(x = 1\). Understanding limits therefore is imperative when verifying continuity analytically in piecewise functions.

Analytical Verification

Having graphically identified potential discontinuities, analytical verification confirms these findings using precise mathematical tools. Continuity at a point requires that the limit of the function as it approaches that point from both sides matches the function’s value. Thus, analytical verification involves:

• Calculating \(\lim_{x \to 1^{-}} \frac{x+1}{x\sqrt{1-x}} = 1\).
• Calculating \(\lim_{x \to 1^{+}} \frac{x^4+1}{x^2} = 2\).
• Noting that \(f(1) = 2\).

With differing left and right limits and a differing function value at \(x = 1\), we confirm analytically that the function is indeed discontinuous at this point, despite being continuous elsewhere. Analytical verification thus cements the insights provided by graphical analysis and the study of limits, serving as a robust method to accurately determine function continuity.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain. They can pose unique challenges when examining their continuity, especially at the points where the expression changes. For our piecewise function:

• The function uses \(f(x) = \frac{x+1}{x\sqrt{1-x}}\) for \(x < 1\).
• It switches to a constant value, \(f(x) = 2\), at \(x = 1\).
• And uses \(f(x) = \frac{x^4+1}{x^2}\) for \(x > 1\).

Key to understanding piecewise functions is studying where transitions occur, such as \(x = 1\). At these critical points, detailed steps are necessary to assess continuity, involving both graphical techniques and precise analytical methods like limits. Grasping piecewise functions and their particular characteristics enriches one’s ability to apply continuity checks across various mathematical scenarios, building a foundation for larger calculus concepts.