Question

In Exercises \(71 - 74 ,\) complete the following tables and state what you believe \(\lim _ { x \rightarrow 0 } f ( x )\) to be. $$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$ $$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\\ \hline \end{array}$$ $$f ( x ) = \frac { 10 ^ { x } - 1 } { x }$$

Short answer

The limit \( \lim _ { x \rightarrow 0 } f ( x )\) is approximately 2.3026.

Step-by-step solution

Calculate the value of \(f(x)\) for negative values

First, take the negative values of \(x\) and substitute them into \(f ( x ) = \frac { 10 ^ { x } - 1 } { x }\). Pay attention to the sign when performing calculations. For example, for \(x = -0.1\), \(f(x) = \frac { 10 ^ { -0.1 } - 1 } { -0.1 } = -0.1053605157\). Do the same for other values of \(x\) in the range.

Calculate the value of \(f(x)\) for positive values

Next, calculate the function values for positive \(x\). For example, for \(x = 0.1\), \(f(x) = \frac { 10 ^ { 0.1 } - 1 } { 0.1 } = 2.302585093\). Continue this for rest of the \(x\) values in this range.

Infer the limit from both sides

From the calculated \(f(x)\) values, observe the trend as \(x\) approaches zero from both sides. As the absolute value of \(x\) decreases, the function value \(f(x)\) seems to approach a certain number from both negative and positive side. This will be the limit of the function as \(x \rightarrow 0\). If the predictions agree, we prove that the function has a limit at \(x = 0\). In our case, it can be concluded that the limit is 2.3026.