Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 19-28, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{t \rightarrow 0} \frac{1-\cos t}{2 t} $$

Step-by-Step Solution

Answer

The limit is 0.25.

Step 1: Understand the Limit Problem

The problem asks us to find the limit of the function \( \frac{1 - \cos t}{2t} \) as \( t \) approaches 0. We will use a graphing calculator to plot the function and numerically evaluate the limit near zero.

Step 2: Numerically Evaluate the Limit

Use a calculator to evaluate the expression \( \frac{1 - \cos t}{2t} \) for values of \( t \) close to 0, such as \(-0.01, -0.001, 0.001, 0.01\). This will help in approximating the limit. Compute each accordingly and observe the trend of the results.

Step 3: Analyze the Values

Calculate \( \frac{1 - \cos(-0.01)}{2(-0.01)} \), \( \frac{1 - \cos(-0.001)}{2(-0.001)} \), \( \frac{1 - \cos(0.001)}{2(0.001)} \), and \( \frac{1 - \cos(0.01)}{2(0.01)} \). You'll find that the values approach approximately \(0.25\).

Step 4: Graph the Function

Plot \( y = \frac{1 - \cos t}{2t} \) on a graphing calculator. Ensure the window is set to view values of \( t \) around 0. Observe the curve as it approaches \( t = 0 \) and confirm that it stabilizes around the apparent limit from numerical evaluations.

Step 5: Draw the Conclusion

Based on the calculations and the graph, you can conclude that the limit of \( \frac{1 - \cos t}{2t} \) as \( t \) approaches 0 is \(0.25\). The graph will further confirm this by showing the function values clustering around \(0.25\) as \( t \) nears zero.

Key Concepts

Limit Evaluation

Limit evaluation is a fundamental aspect of calculus that helps us understand the behavior of functions as they approach specific points. In the problem, we need to find the limit of the function \( \frac{1 - \cos t}{2t} \) as \( t \) approaches 0. One common technique for evaluating such a limit is to use numerical approximation techniques.

• Start by selecting values very close to the point of interest, in this case, numbers like \(-0.01, -0.001, 0.001, \) and \( 0.01 \).
• Plug these values into the limit expression to compute the values at these points.
• The goal is to observe the pattern or trend of the calculated values. As we calculate \( \frac{1-\cos t}{2t} \) for values close to zero, we notice that these outputs cluster towards 0.25.

This pattern is crucial in evaluating limits—it indicates the value that the function is approaching, which is precisely the limit value.

Graphical Analysis

Graphical analysis involves visually examining the function to better understand its behavior as it nears the limit. To achieve this, a graphing calculator or software can be employed. Follow these steps:

• First, plot the graph of \( y = \frac{1 - \cos t}{2t} \) over a domain that includes \( t \) values around zero.
• Adjust your viewing window to focus closely on the points around \( t = 0 \), where the limit is evaluated.
• Observe how the graph behaves as \( t \) comes closer to zero. The function's output values should settle around the y-value corresponding to the estimated limit, in this case, 0.25.

By using such graphical analyses, students can visually confirm numerical findings and develop an intuition about how functions behave near specific points. This helps solidify understanding by combining both visual and arithmetic evidence.

Trigonometric Limits

Understanding trigonometric limits is essential when dealing with limits that involve trigonometric functions, such as sine, cosine, or tangent. The given function \( \frac{1 - \cos t}{2t} \) involves the cosine function, which naturally links to specific trigonometric limits known through calculus principles.

• One related trigonometric limit is \( \lim_{t \to 0} \frac{\sin t}{t} = 1 \), which aids in understanding similar limits involving \( \cos t \).
• The expression \( 1 - \cos t \) exhibits behavior at small values of \( t \) that can be related to such known limits.
• Using L'Hôpital's Rule can also be an alternative for evaluating limits where direct substitution results in an indeterminate form. This rule involves differentiating the numerator and denominator separately and then taking the limit.

Learning about these fundamental trigonometric limits and rules like L'Hôpital’s enhances one's ability to solve complex trigonometric limit problems fluently and effectively. They provide a toolkit for simplifying otherwise intricate problems.