Question

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

Short 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-by-step solution

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 \(x = -0.1, -0.01, -0.001,-0.0001\).

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 \(x = 0.1, 0.01, 0.001, 0.0001\).

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.