Question

In Exercises \(71 - 74 ,\) complete the following tables and state what you believe \(\lim _ { x \rightarrow 0 } f ( x )\) to be. $$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$ $$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\\ \hline \end{array}$$ $$f ( x ) = x \sin \frac { 1 } { x }$$

Short answer

The limit of the function \(f(x) = x \sin (1/x)\) as \(x\) approaches 0 is 0.

Step-by-step solution

Fill in the Tables with Given Values

Calculate the function value for each provided \(x\) value in the table. For the first given \(x\) value, -0.1, evaluate the function as \(f(x) = -0.1 \sin (1/-0.1)\). Repeat this calculation for remaining \(x\) values in the tables.

Analyze the Table Values

After calculating and filling the values into the tables, analyze them to understand how the function behaves as \(x\) approaches 0 from both left and right. You will notice that as \(x\) gets closer to 0, the function values swings between -1 and 1, with swinging amplitude gets smaller.

Approximate the Limit

Based on the filled tables and observation in step 2, approximate the limit of \(f(x)\) as \(x\) approaches 0. Since the function swings between a decreasing maximum and minimum and those maximum and minimum are getting closer to 0, we can say that \(\lim _ { x \rightarrow 0 } f ( x )\) approaches 0.

Key concepts

Calculating Limits

When it comes to calculating limits, the process involves determining what value a function approaches as the input approaches a certain point. This concept is fundamental in calculus, providing insight into a function's behavior around specific points, even points where the function might not be clearly defined.

A limit calculation can be approached algebraically by manipulating the function towards simplicity, graphically by looking at a plot and predicting the trend, or, as in the exercise, numerically by evaluating the function at values closer and closer to the point of interest. For instance, by computing values of a function for inputs like 0.1, 0.01, and so on, one can observe patterns that lead to a logical conclusion about the limit.

Continuity and Limits

The concepts of continuity and limits are tightly interwoven. A function is continuous at a point if the limit as you approach that point equals the function's value at that point. If there's a discrepancy, the function is not continuous there.

Continuity has essential implications because if a function is continuous at a given point, you can confidently say that small changes in input will result in small changes in the output. When the exercise prompts you to fill out a table of values of a function as the inputs get infinitesimally close to zero, what's being evaluated is precisely this concept of continuity, specifically whether or not the limit exists at that point.

Sine Function Behavior

The sine function's behavior is an excellent example of how functions can oscillate between values. For inputs very close to zero, \( sin(1/x) \) exhibits very rapid oscillations.

The sine function has a well-known pattern; it swings between -1 and 1 over regular intervals. When multiplied by \( x \) as in \( x \sin(1/x) \) from the exercise, these oscillations occur between \( -x \) and \( x \) which decrease in amplitude as \( x \) approaches zero. This shrinking swing is what you observe in the tables when calculating the function at smaller and smaller negative and positive values of \( x \) and is a manifestation of the sine function’s behavior within a larger expression.

Approaching Limits

The concept of approaching limits hinges on analyzing a function's value as the input 'approaches' a certain point, from both the left and the right. It isn't about the value at the point precisely, but rather what value the function tends towards as it gets infinitely close to that point.

In many exercises, like the one provided, this 'approach' is represented by a table of increasingly fine values, showing the trend. Here, as \( x \) gets closer to zero, the swinging behavior of \( f(x) \) decreases in range, indicating that the values of \( f(x) \) are 'zeroing in' on a particular number. The exercise instructs students to look at these trends and estimate the limit based on this behavior of approaching ever closer but never quite reaching until the actual limit is calculated or approximated.