Problem 73 - 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. $$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$ $$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\\ \hline \end{array}$$ $$f ( x ) = \frac { 10 ^ { x } - 1 } { x }$$

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.

Key Concepts

Function Evaluation

In order to understand limits, it's crucial to first know how to evaluate a function. Function evaluation involves substituting different values into the function to determine its output. Consider the given function: \(f(x) = \frac{10^x - 1}{x}\). Here's how to evaluate it:

• Choose a value of \(x\).
• Substitute this value into the expression for \(f(x)\).
• Use basic arithmetic and exponentiation to compute the result.

Let's try evaluating \(f(x)\) for a negative value like \(x = -0.1\). After substituting, calculate:
\(f(-0.1) = \frac{10^{-0.1} - 1}{-0.1}\), which results in \(-0.1053605157\). Applying the same method to positive values, such as \(x = 0.1\), we find \(f(x)\) equates to \(2.302585093\). Evaluating functions systematically is the first step toward understanding limits.

Limit Calculation

Calculating the limit of a function like \(f(x) = \frac{10^x - 1}{x}\) is a fundamental concept in calculus. It involves determining what value \(f(x)\) approaches as \(x\) gets closer and closer to a particular point, in this case, zero. This is done as follows:

• Estimate the function values as \(x\) approaches zero from both the negative and positive sides.
• Observe any patterns or convergence in the values.

The tables provided in the exercise help visualize this by displaying \(f(x)\) for increasingly small values of \(x\). From the calculations:

• Negative approach: \(-0.1, -0.01, -0.001\)
• Positive approach: \(0.1, 0.01, 0.001\)

As \(x\) nears zero, \(f(x)\) seems to converge toward the same number from both sides, leading to the conclusion that the limit is \(2.3026\). Determining limits helps predict the behavior of functions near critical points where direct evaluation is not possible.

Numerical Approach

A numerical approach to finding limits involves using tables or a set of values close to the specified point. It's an effective way to estimate limits when an algebraic method is not directly apparent or practical. Here's how it works:

• Select values of \(x\) incrementally closer to the point of interest, such as zero.
• Evaluate \(f(x)\) for these values.
• Record the function outputs to determine the trend.

This approach is especially valuable for complex functions or to supplement analytical methods. In the exercise, by examining values such as \(x = 0.1, 0.01, -0.01, -0.001\),
we note that \(f(x)\) heads toward \(2.3026\) on both sides. Although precise calculation of limits can involve intricate formulas,
the numerical approach provides a confirmatory way to view function behavior,
bridging theoretical understanding with practical observation.