Problem 74 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(71 - 74 ,\) complete the following tables and state what you believe \(\lim _ { x \rightarrow 0 } f ( x )\) to be. $$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$ $$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\\ \hline \end{array}$$ $$f ( x ) = x \sin ( \ln | x | )$$

Step-by-Step Solution

Answer

To find the accurate answer of the limit, calculations of function values for the given negative and positive proximities of zero should be performed. Depending on the trend of these calculations, a conclusion about the limit as \(x\) tends to zero can be made.

Step 1: Calculate the function values for negative proximities of zero

The first table asks for values of \(f(x)\) at negative points, that is when \(x < 0\). This means we have to compute \(f(x)\) for \(x = -0.1, -0.01, -0.001,-0.0001\).

Step 2: Calculate the function values for positive proximities of zero

The second table is requesting function values for positive \(x\), hence, calculate the values of \(f(x)\) for \(x = 0.1, 0.01, 0.001, 0.0001\).

Step 3: Deduce the limit value as x tends to zero

After obtaining all the values in both cases, it's possible to make a deduction about the limit of the function as \(x\) tends to zero. The limit of a function as \(x\) tends to a particular value is the value that the function approaches as \(x\) gets infinitely close to that value. This should be done by looking at the trend of the function values as \(x\) gets close to zero from both positive and negative directions. If the function values approach the same number from both directions, then that number is the limit as \(x\) tends to zero.

Key Concepts

Evaluating Limits

When we're evaluating limits, especially in calculus, we are trying to find out what value a function approaches as the input, or in this case, as the variable \(x\) comes closer and closer to a certain point. In this exercise, the focus is on figuring out what happens to the function \(f(x) = x \sin(\ln |x|)\) as \(x\) gets closer to zero.

Evaluating limits involves a few critical steps. First, you determine the function values at points increasingly near to the point of interest, zero in this instance. This includes looking from both the left (negative values of \(x\)) and the right (positive values of \(x\)). This two-sided observation is fundamental in ensuring the accuracy of your limit deduction.

To precisely calculate these function values, you essentially plug in values into the function and evaluate the respective outcomes.

• Calculating \(f(x)\) for negative values like \(-0.1, -0.01, \) etc.
• Calculating \(f(x)\) for positive values such as \(0.1, 0.01, \), and so on.

The essence of this approach is to identify patterns or trends in the function values as \(x\) approaches the number zero.

Behavior of Functions around Zero

Understanding how functions behave around zero is crucial in mathematics because it often reveals key insights about the function's characteristics. With \(f(x) = x \sin(\ln |x|)\), we must examine how it acts as \(x\) becomes very small, approaching zero from both sides.

This means we need to observe the values plotted when \(x\) is a tiny positive number and a tiny negative number. What we typically look for is whether these numbers get bigger, smaller, or stay about the same as they converge on zero. This kind of behavior tells us a lot about the nature of \(f(x)\), especially whether it is trying to stabilize around a particular value or if it is exhibiting more erratic behavior.

Because the amplitude of \(\sin(\ln |x|)\) is always between -1 and 1, observing the changes as \(x\) becomes tiny helps us understand if the behavior of the function is consistent and if it might settle on a value as it nears zero.

• Are these values approaching a limit?
• Are they oscillating between the same bounds?
• Or, are they diverging completely?

Recognizing these trends forms the basis for determining what truly happens to \(f(x)\) behaviorally as \(x\) tends to zero.

Calculating Function Values

The calculation of function values underlies all of limit evaluation. Here, it means calculating the precise outcome of \(f(x) = x \sin(\ln |x|)\) as \(x\) nears zero for each designated point. The approach simplifies to substituting each individual \(x\) value into the function and working through the arithmetic to find \(f(x)\).

To carry this out seamlessly, remember to handle the expression \(\ln |x|\) by taking absolute values wherever \(x\) might initially appear negative. This guarantees that you're evaluating the logarithm of a positive number, as expected in this context.

• Calculate \(f(x)\) at each step: \(f(-0.1), f(-0.01), f(-0.001), etc.\)
• Also compute for positive \(x\) like \(f(0.1), f(0.01), f(0.001), etc.\)

Comparing these function outputs allows for the deduction of a trend or limit, assisting us in pinpointing what \(f(x)\) converges toward as \(x\) approaches zero. It's this precise method of calculation that fuels our understanding of limits and function behaviors within calculus.