Problem 56 - Calculus Early Transcendentals

Question

It can be shown that \(1-\frac{x^{2}}{2} \leq \cos x \leq 1,\) for \(x\) near 0. a. Illustrate these inequalities with a graph. b. Use these inequalities to evaluate \(\lim _{x \rightarrow 0} \cos x\).

Step-by-Step Solution

Answer

Question: Determine the limit of cosine x as x approaches 0 using the given inequality \(1-\frac{x^{2}}{2} \leq \cos x \leq 1\). Answer: The limit of cosine x as x approaches 0 is 1.

Step 1: Part a: Graphing the inequalities

To graph the inequalities, we need to visualize the functions \(y_1 = 1-\frac{x^2}{2}\), \(y_2 = \cos x\), and \(y_3 = 1\). Create a plot with x-axis ranging from \(-2\) to \(2\), so it is clear what happens when x is near 0. 1. Plot the function \(y_1 = 1 - \frac{x^2}{2}\) on the same graph. This is a parabolic function opening downwards with a vertex at (0,1). 2. Plot the cosine function \(y_2 = \cos x\) on the same graph. This is a periodic function crossing the y-axis at 1. 3. Plot the constant function \(y_3 = 1\) on the same graph. This is a horizontal line along y=1. Notice that for x near 0, the inequality \(1-\frac{x^{2}}{2} \leq \cos x \leq 1\) holds true, as the cosine function is always between the two_ other curves.

Step 2: Part b: Evaluate the limit of cosine x as x approaches 0.

To solve this, we will utilize the Squeeze Theorem, which states that if the function \(g(x) \leq f(x) \leq h(x)\) for all x near a point, then: $$\lim _{x \rightarrow a} g(x) \leq \lim _{x \rightarrow a} f(x) \leq \lim _{x \rightarrow a} h(x)$$ In our case, we have: $$1-\frac{x^{2}}{2} \leq \cos x \leq 1$$ Now, let's calculate the limits: $$\lim _{x \rightarrow 0} (1-\frac{x^2}{2}) = 1 - \frac{0^2}{2} = 1$$ $$\lim _{x \rightarrow 0} (1) = 1$$ So by the Squeeze Theorem, since: $$\lim _{x \rightarrow 0} (1-\frac{x^2}{2}) = \lim _{x \rightarrow 0} (1) = 1$$ We can conclude that the limit of cosine x as x approaches 0 is: $$\lim _{x \rightarrow 0} \cos x = 1$$

Key Concepts

Limits in Calculus

Limits are a fundamental concept in calculus that help us understand the behavior of functions as they approach a specific point. Instead of calculating a function's value at a point directly, we look at what happens as we get closer to that point. For instance, when we say \(\lim_{x \to 0} \cos x = 1\), we're interested in what happens to the cosine function as x gets closer and closer to 0.

This idea is vital for defining some of the more complex operations in calculus, like derivatives and integrals. Limits help us handle situations where direct evaluation might be difficult or impossible. It's like predicting where a runner will be in a race without actually stopping them at a given moment. By looking at limits, we get to understand more about the function’s overall behavior.

Graphing Inequalities

Graphing inequalities involves plotting multiple functions and examining their relative positions. In our exercise, we have three functions: \(y_1 = 1 - \frac{x^2}{2}\), \(y_2 = \cos x\), and \(y_3 = 1\). Each functions serves as a boundary or condition.

To illustrate the inequality \(1-\frac{x^2}{2} \leq \cos x \leq 1\), we place all functions on a graph:

• \(y_1 = 1-\frac{x^2}{2}\), a parabola upside down with its vertex at the top.


• \(y_2 = \cos x\), a wave-like curve that repeats and is centered around y = 1.


• \(y_3 = 1\), a straight horizontal line.

For x near 0, observing these functions visually shows the cosine curve wedged between the parabola and the straight line, thus satisfying the inequality. This graphical representation is a powerful way to see mathematical relationships, especially in calculus.

Cosine Function

The cosine function \(\cos x\) is one of the basic trigonometric functions. It flutters between -1 and 1 in a wave-like pattern. Starting at \(\cos 0 = 1\), it repeats every \(2\pi\) units. These repeated patterns make it a periodic function, which means \(\cos(x + 2\pi) = \cos x\).

The cosine function connects to various concepts in calculus:

• It is continuous, meaning there are no breaks or holes in its graph.


• It is smooth and differentiable, which means we can calculate its slope at any point.


• Close to zero, \(\cos x\) can be approximated using a quadratic, which is why \(y_1 = 1 - \frac{x^2}{2}\) serves as a close representation near that point.

Because of its unique properties, the cosine function is critical to understanding waves, oscillations, and other applications where periodicity is a factor.

Approaching a Limit

Approaching a limit means getting closer and closer to a particular value as x tends to a specific point, without necessarily reaching that point. This is like walking closer to a goal line without crossing it. Calculus uses this idea to find limits both from the left and the right.

In the context of the Squeeze Theorem, which is a way to "trap" a function's limit using bounds, approaching the limit is visualized by getting near a desired value through comparison. When the function \(\cos x\) is squeezed by \(1 - \frac{x^2}{2}\) and \(1\) as x gets close to zero, we see they both converge towards 1.

This approach solidifies our conclusion and helps prove that \(\lim_{x \to 0} \cos x = 1\). By checking values trending from either side of 0, one confirms both limits align, making approaching limits essential for proving bounds and understanding function behavior in challenging cases.