Problem 27 - Precalculus

Question

Use a table of values to evaluate each function as \(x\) approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation. $$s(x)=\frac{2 \cos x-2}{x}, x \rightarrow 0$$

Step-by-Step Solution

Answer

As \(x\) approaches 0, \(s(x)\) approaches 0. Thus, \(\lim_{{x \to 0}} \frac{2 \cos x - 2}{x} = 0\).

Step 1: Understand the Function

We want to evaluate the function \(s(x) = \frac{2 \cos x - 2}{x}\) as \(x\) approaches 0. This function has the form of a difference quotient, a type often seen in limits.

Step 2: Create a Table of Values

Let's choose values of \(x\) that are close to 0 and calculate \(s(x)\). Use both positive and negative values for \(x\) to see the trend from both directions. For example, choose \(-0.1, -0.01, 0.01, 0.1\).

Step 3: Calculate Function Values

Compute \(s(x)\) for each chosen \(x\):- \(x = -0.1\), \(s(-0.1) = \frac{2 \cos(-0.1) - 2}{-0.1}\).- \(x = -0.01\), \(s(-0.01) = \frac{2 \cos(-0.01) - 2}{-0.01}\).- \(x = 0.01\), \(s(0.01) = \frac{2 \cos(0.01) - 2}{0.01}\).- \(x = 0.1\), \(s(0.1) = \frac{2 \cos(0.1) - 2}{0.1}\).Calculate these values precisely.

Step 4: Analyze the Table

Observe the calculated values of \(s(x)\). If the values converge towards a particular number as \(x\) gets closer to 0, this is the limiting value of the function.

Step 5: Conclude with Limit Notation

If the function values from Step 4 approach 0, write the relationship in words and using limit notation. We can say: "As \(x\) approaches 0, the function \(s(x)\) approaches 0." The limit statement is:\[ \lim_{{x \to 0}} \frac{2 \cos x - 2}{x} = 0 \]

Key Concepts

Difference Quotient

The concept of the difference quotient forms the backbone of calculus, particularly when we deal with evaluating the behavior of functions as they approach specific points. In our exercise, the function \( s(x) = \frac{2 \cos x - 2}{x} \) is structured like a difference quotient.

The difference quotient is often expressed as \( \frac{f(x+h) - f(x)}{h} \), which represents the average rate of change of a function over a small interval \(h\). In the context of limits, it helps us understand instantaneous rates of change.

For this function, the numerator \(2 \cos x - 2\) suggests the change is based on modifying the \(\cos\) function, which we then evaluate as \(x\) approaches 0.

Trigonometric Functions

Trigonometric functions like \(\cos x\) play a key role in many limit problems. In our exercise, the behavior of \(\cos x\) as \(x\) gets closer to zero affects the limit of the whole function \(s(x) = \frac{2 \cos x - 2}{x}\).

Here, it's important to recall that \(\cos(0) = 1\). When you evaluate \(2 \cos x - 2\), it simplifies to 0 when \(x = 0\). This outcome influences the behavior of \(s(x)\) significantly as \(x\) tends to zero.

When using tables for approximation, it's useful to observe how \(\cos x\) remains close to 1 for small values of \(x\). This helps in predicting the limiting value of \( \frac{2 \cos x - 2}{x} \).

Limit Notation

When dealing with limits in mathematics, limit notation is essential for conveying precise mathematical relationships. In this specific problem, we determined the behavior of \(s(x) = \frac{2 \cos x - 2}{x}\).

Write in words, "As \(x\) approaches 0, the function \(s(x)\) approaches 0." In limit notation, this entire idea is concisely expressed as:

• \( \lim_{{x \to 0}} \frac{2 \cos x - 2}{x} = 0 \)

This notation captures how the values of \(s(x)\) become arbitrarily close to 0 when \(x\) is close to 0, a crucial aspect of analyzing functions in calculus.

Approaching Values

In calculus, "approaching values" is a fundamental idea that describes how a function behaves as its input gets closer to a certain number. For the function \(s(x) = \frac{2 \cos x - 2}{x}\), we observe what happens as \(x\) nears 0 from both positive and negative sides.

When setting up a table, we use values such as \(-0.1, -0.01, 0.01, 0.1\). By calculating \(s(x)\) for these values, we notice a trend.

This examination reveals that as \(x\) moves closer to 0, the output values of \(s(x)\) converge towards 0, confirming the limit. Hence, in limit problems, understanding the behavior of a function as it "approaches" certain values is key in determining the limiting behavior.