Problem 41 - Contemporary Precalculus

Question

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow 0}(x \ln |x|)$$

Step-by-Step Solution

Answer

Answer: The limit of the function \(f(x) = x \ln|x|\) as \(x\) approaches 0 is 0. However, this limit does not equal to \(f(c)\) since the function is undefined for \(x = 0\).

Step 1: Investigate the function's behavior near the limit point

To find the limit as \(x\) approaches 0, we need to investigate the behavior of \(f(x) = x \ln|x|\) near 0. Since the natural logarithm is undefined for negative values of its argument, we need to consider both left and right limits.

Step 2: Observe the function graphically

You can easily use a graphing calculator or an online graphing tool to observe the behavior of the function near x=0. As you can see in the graph, the function seems to approach some value as x approaches 0 from both left and right side.

Step 3: Use numerical means to find the limit

To find the limit numerically, we can create a table with decreasingly smaller values for x approaching 0 from both sides and compute the corresponding y values using the function. We can create the table as follows: | x | y(x) | |---|------| | 0.1 | -0.331 | | 0.01 | -0.0461 | | 0.001 | -0.00691 | |-0.1 | -0.331 | |-0.01 | -0.0461 | |-0.001 | -0.00691 | Looking at the y values in the table, it appears that y(x) approaches 0 as x approaches 0.

Step 4: Conclusion

Based on the graphical and numerical investigation, we can conclude that: $$\lim _{x \rightarrow 0}(x \ln |x|)=0$$ Since the limit exists, let's determine whether it is equal to \(f(c)\): The given function is \(f(x) = x \ln|x|\). The value of f(c) when c = 0 is undefined. Since f(c) is undefined and the limit as x approaches 0 exists and equals to 0, we can conclude that: $$\lim _{x \rightarrow 0}(x \ln |x|) \neq f(c)$$

Key Concepts

Numerical Approximation of Limits

Understanding the concept of the limit of a function is fundamental in calculus and it pertains to what value a function approaches as the input gets close to a certain point. When we cannot find the limit analytically, numerical approximation comes into play.

Numerical approximation involves calculating the function's value at various points that are extremely close to the point of interest, from both directions if necessary. This method gives us a sequence of values that should get closer and closer to the actual limit. For instance, to approximate the limit of the function \( f(x) = x \ln|x| \) as \( x \) approaches 0, we calculate the function's value at points like 0.1, 0.01, 0.001, and so on, and observe their behavior.

As shown in the original exercise, by creating a table of values approaching zero, we can see a trend. The values of \( y \) become progressively smaller and appear to approach zero, which suggests that the limit of the function as \( x \) approaches 0 is 0.

Graphical Analysis of Limits

Another method to understand the behavior of functions as they approach a particular value is graphical analysis. Graphs give a visual representation of how a function behaves and what it tends towards as input values approach a particular point.

For the function \( f(x) = x \ln|x| \), graphing it would reveal how the function's output changes around \( x = 0 \). Despite the fact that the natural logarithm, \( \ln|x| \), tends towards negative infinity as \( x \) approaches 0, the product \( x \ln|x| \) actually tends to 0. This outcome might seem counterintuitive, which is why graphing is such a helpful tool. It allows us to see the overall trend even when the function's behavior is complex. As evidenced within the problem's provided solution, the graphical analysis shows the function approaching a value, hinting at the existence of a limit.

Natural Logarithm Properties

To fully understand the exercise, one must grasp certain properties of the natural logarithm. The natural logarithm, denoted as \( \ln(x) \), is the inverse function of the exponential function \( e^x \), it's defined only for positive real numbers. A key property is that \( \ln(1) = 0 \), and as \( x \) approaches 0 from the right, \( \ln(x) \) approaches negative infinity.

Moreover, the natural logarithm is continuous and grows without bound as \( x \) increases, which means it gets larger and larger as \( x \) gets larger, but does so at a decreasing rate. For the function at hand, \( x \ln|x| \) leverages the property that \( \ln(x) \) is negative for \( 0 < x < 1 \), which helps explain why the limit of \( x \ln|x| \) as \( x \) approaches 0 is 0, since the \( x \) factor mitigates the negative infinity behavior of the \( \ln|x| \) part.