Problem 34 - Precalculus: Mathematics for Calculus

Question

Graphing Calculator Pitfalls (a) Evaluate $$ h(x)=\frac{\tan x-x}{x^{3}} $$ for \(x=1,0.5,0.1,0.05,0.01,\) and 0.005 (b) Guess the value of \(\lim _{x \rightarrow 0} \frac{\tan x-x}{x^{3}}\) (c) Evaluate \(h(x)\) for successively smaller values of \(x\) until you finally get a value of 0 for \(h(x) .\) Are you still confident that your guess in part (b) is correct? Explain why you eventually got a value of 0 for \(h(x)\) (d) Graph the function \(h\) in the viewing rectangle \([-1,1]\) by \([0,1] .\) Then zoom in toward the point where the graph crosses the \(y\) -axis to estimate the limit of \(h(x)\) as \(x\) approaches \(0 .\) Continue to zoom in until you observe distortions in the graph of \(h .\) Compare with your results in part (c).

Step-by-Step Solution

Answer

The limit is 1 as \(x\) approaches 0. The calculator may later show 0 due to rounding errors.

Step 1: Substitute Values into h(x)

Begin by substituting each given value of \(x\) into the function \(h(x) = \frac{\tan x - x}{x^3}\) to evaluate \(h(x)\) for \(x = 1, 0.5, 0.1, 0.05, 0.01,\) and 0.005. Make sure to calculate \(\tan x\) in radians.

Step 2: Calculate h(x) for Given Values

\(h(1) = \frac{\tan(1) - 1}{1^3} \approx 0.5571\)\(h(0.5) = \frac{\tan(0.5) - 0.5}{0.5^3} \approx 0.6438\)\(h(0.1) = \frac{\tan(0.1) - 0.1}{0.1^3} \approx 0.9983\)\(h(0.05) = \frac{\tan(0.05) - 0.05}{0.05^3} \approx 0.9997\)\(h(0.01) = \frac{\tan(0.01) - 0.01}{0.01^3} \approx 1.0000\)\(h(0.005) = \frac{\tan(0.005) - 0.005}{0.005^3} \approx 1.0000\)

Step 3: Estimate the Limit

The values of \(h(x)\) as \(x\) approaches 0 suggest that \(\lim_{x \rightarrow 0} \frac{\tan x - x}{x^3} = 1\). This is based on the numerical results which converge to 1 as \(x\) becomes smaller.

Step 4: Evaluation with Smaller x-Values

Continue evaluating \(h(x)\) for smaller \(x\) values, such as \(x = 0.0001\,\, and \ ,00001\) to observe whether \(h(x)\) remains close to 1. At some point, due to precision limits of the calculator, \(h(x)\) may inaccurately evaluate to 0.

Step 5: Analyze the Calculator's Precision

If a very small value of \(x\) results in \(h(x) = 0\), it is likely due to the calculator rounding errors and limited precision, not because the true limit is 0. The theoretical calculations support that the limit is 1.

Step 6: Graph the Function

Plot the graph of \(h(x)\) over \([-1, 1]\) by \([0, 1]\). Notice how the graph behaves around \(x = 0\) under this viewing window.

Step 7: Zoom in on the y-axis

Zoom in towards the y-axis on the point where the graph appears to cross it as \(x\) approaches 0. Continue to zoom in closer to observe any changes or unexpected behavior in the graph, comparing the visual result to the expected limit of 1.

Key Concepts

Limit Evaluation

Limit evaluation is a fundamental concept in calculus, and it involves determining the value a function approaches as the input approaches a certain point. In this exercise, we're looking at the function \( h(x) = \frac{\tan x - x}{x^3} \). Our goal is to find what happens to \( h(x) \) as \( x \) becomes very close to zero.

We begin by substituting decreasing values of \( x \) into \( h(x) \), such as \(1, 0.5, 0.1\), and so on. These substitutions give us an idea of the behavior of the function near the point of interest, \( x = 0 \). As we see in the calculations, \( h(x) \) seems to be converging to 1 as \( x \) gets smaller. This suggests that the limit of \( h(x) \) as \( x \) approaches zero is 1.

Evaluating limits analytically often requires differentiating or applying special techniques, but here, numerical evaluation provides strong evidence. This method is insightful, especially when the formula is complex or an analytical approach seems less straightforward.

Numerical Approximation

Numerical approximation is using numbers to find an estimated solution when an exact solution is difficult to achieve. In this case, computing the expression \( h(x) \) for various values helps us approximate the limit of a function.

Key points to keep in mind include:

• Be consistent in using units; radians for trigonometric functions is standard in calculus.
• Understand that smaller \( x \) values offer a clearer picture of the behavior around the point of interest.

We calculate values like \( h(0.01) \approx 1 \). These results approximate the behavior as \( x \) approaches zero, suggesting that \( h(x) \) approaches 1, confirming our manual limit estimation.

However, students should be aware that numerical approximation can be limited by the tool used. It offers a great overview and estimation that leads to the solution but may not provide exactness over tiny numeric fields.

Rounding Errors

Rounding errors occur as a side effect of representing numbers with finite precision in calculators or computers. When evaluating \( h(x) \) for very small values of \( x \), our results were more precise at first but eventually displayed \( h(x) = 0 \) instead of a value close to 1.

This zero represents the rounding errors and limitations of digital devices, not a mathematical truth. This happens because, as calculations involve numbers getting infinitesimally small, the finite memory of a machine impacts accuracy.

To handle rounding errors:

• Be cautious with results for extremely small numbers.
• Know that unexpected outcomes, like zero where you expect a non-zero value, can occur.
• Use rounding rules wisely and double-check unusual outputs with another calculation method.

Recognizing these errors helps prevent misinterpretation when using graphing calculators or similar digital tools.

Graphing Techniques

Graphing is a visual way to understand the behavior of functions. It can expose trends and potential issues that are not evident in raw data alone. For \( h(x) \), graphing helps visualize how \( h(x) \) behaves as \( x \) approaches 0.

When plotting the function on a graphing calculator:

• Choose appropriate scales on both axes, like \([-1, 1]\) for \( x \) and \([0, 1] \) for \( y \), to focus on the point of interest.
• Use the zoom feature to get closer to suspected intersection points or points where the behavior is changing. This helps in understanding the function around tiny neighborhood points such as where \( x = 0 \).

With each zoom, pay attention to any distortion; it could indicate issues, like rounding errors, and reveal more about deviations from expected behavior. By carefully observing the shape of the graph, students can correlate the visual data with their numerical calculations to enhance understanding.