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Chapter 1: Problem 19

(a) By graphing the function \(f(x)=(\cos 2 x-\cos x) / x^{2}\) and zooming in toward the point where the graph crosses the \(y\) -axis, estimate the value of \(\lim _{x \rightarrow 0} f(x)\) (b) Check your answer in part (a) by evaluating \(f(x)\) for values of \(x\) that approach \(0 .\)

Short Answer

Expert verified
The limit \( \lim_{x \to 0} \frac{\cos 2x - \cos x}{x^2} \) is approximately \(-3\).

Step by step solution

01

Understand the Function

The function given is \( f(x) = \frac{\cos 2x - \cos x}{x^2} \). We need to estimate \( \lim_{x \to 0} f(x) \) by graphing and evaluating it near \( x = 0 \).
02

Graph the Function

Using a graphing tool or graphing calculator, plot the function \( f(x) = \frac{\cos 2x - \cos x}{x^2} \). Zoom in on the region close to \( x = 0 \). Observe the behavior of the function as \( x \) approaches zero from both sides.
03

Estimate from the Graph

From the graph, you should see that as you zoom closer to \( x = 0 \), the values of \( f(x) \) converge towards a particular value. This is the graphical estimation of the limit.
04

Choose Values of x

Select small positive and negative values approaching 0, such as \( x = 0.1, 0.01, -0.1, -0.01 \), and compute \( f(x) \) for each value to numerically observe the limit.
05

Calculate f(x) for Chosen Values

Evaluate \( f(x) = \frac{\cos 2x - \cos x}{x^2} \) for the selected values. Example calculations: - For \( x = 0.1 \), calculate \( f(0.1) \).- For \( x = 0.01 \), calculate \( f(0.01) \).- Similarly, calculate for negative values \( x = -0.1 \) and \( x = -0.01 \).
06

Analyze Numerical Results

Compare the calculated values of \( f(x) \). If they approach a single number as \( x \) approaches zero, that number is the estimated limit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphical estimation of limits
Graphical estimation is a powerful technique to understand the behavior of functions as they approach a particular value. To estimate limits graphically, we typically plot the function and closely observe its behavior as we hone in on the point of interest. In the case of our function, \( f(x) = \frac{\cos 2x - \cos x}{x^2} \), the point of interest is \( x = 0 \).
Here are steps to ensure precise estimation:
  • Plot the function using graphing software or a calculator.
  • Ensure your graph is accurate by using a fine scale, especially near \( x = 0 \).
  • Zoom in repeatedly, focusing on the region around \( x = 0 \), and observe the \( y \)-value towards which the function converges.
By doing these steps, you should notice that as \( x \) gets closer to zero from both the positive and negative directions, the values the graph approaches give a good visual idea of the limit. This graphical convergence is a visual representation of the mathematical limit.
Numerical methods for limits
Numerical methods are a great way to support the graphical estimation of limits. These methods involve calculating the value of the function at points that progressively get closer to the value we're interested in鈥攊n this instance, \( x = 0 \).
For the function \( f(x) = \frac{\cos 2x - \cos x}{x^2} \), we use specific values of \( x \):
  • Start with small increments such as \( x = 0.1 \) and \( x = -0.1 \).
  • Progress to smaller values, like \( x = 0.01 \) and \( x = -0.01 \).
Calculate \( f(x) \) for each chosen \( x \) value. For instance:
  • \( f(0.1) \)
  • \( f(0.01) \)
  • \( f(-0.1) \)
  • \( f(-0.01) \)
As you compute these values, observe the trend. If all evaluations show the function approaching a specific number as \( x \) gets closer to zero, this trend numerically verifies the function鈥檚 limit.
Cosine function behavior
Understanding the behavior of the cosine function is key to evaluating limits involving trigonometric expressions. The cosine function, \( \cos(x) \), oscillates between -1 and 1, and has a periodic nature that repeats every \( 2\pi \).
In the given function, \( f(x) = \frac{\cos 2x - \cos x}{x^2} \), the subtraction \( \cos 2x - \cos x \) requires careful examination of frequency and amplitude. Here鈥檚 what to note:
  • \( \cos 2x \) oscillates twice as fast as \( \cos x \). This difference can lead to rapid changes in \( \cos 2x - \cos x \) as \( x \) changes slightly.
  • Near \( x = 0 \), both \( \cos 2x \) and \( \cos x \) approach 1, but \( \cos 2x - \cos x \) would present small values due to subtle differences.
Understanding these oscillations helps us break down why the function behaves the way it does near our point of interest, making sense of the computations required for estimating limits.

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Most popular questions from this chapter

Suppose that a function \(f\) is continuous on \([0,1]\) except 0.25 and that \(f(0)=1\) and \(f(1)=3 .\) Let \(N=2 .\) Sketch two possible graphs of \(f,\) one showing that \(f\) might not satisfy the conclusion of the Intermediate Value Theorem and one showing that \(f\) might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesn't satisfy the hypothesis).

Evaluate the limit, if it exists. $$\lim _{u \rightarrow 2} \frac{\sqrt{4 u+1}-3}{u-2}$$

\(29-30=\) Show that \(f\) is continuous on \((-\infty, \infty)\) $$f(x)=\left\\{\begin{array}{ll}{\sin x} & {\text { if } x<\pi / 4} \\ {\cos x} & {\text { if } x \geqslant \pi / 4}\end{array}\right.$$

How close to \(-3\) do we have to take \(x\) so that $$\frac{1}{(x+3)^{4}}>10,000$$

Evaluate the limit, if it exists. $$\lim _{x \rightarrow 16} \frac{4-\sqrt{x}}{16 x-x^{2}}$$
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