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Chapter 11: Problem 19

Calculate. $$\lim _{x \rightarrow \infty}\left(\cos \frac{1}{x}\right)^{2}$$

Short Answer

Expert verified
In summary, the limit of \(\left(\cos \frac{1}{x}\right)^{2}\) as \(x\) approaches infinity is \(\mathbf{1}\).

Step by step solution

01

Identify the given expression

(\(\cos \frac{1}{x}\))^{2} as x approaches infinity.
02

Analyze the cosine function at the limit

We know that \(\cos(0)\) is 1, and as x approaches infinity, the value of \(\frac{1}{x}\) approaches 0. Therefore, we can rewrite the expression as \(\cos(0)\)^{2} at the limit.
03

Substitute and compute the limit

Replace \(\frac{1}{x}\) with 0 in the given expression: (\(\cos 0\))^{2}. Compute \(\cos(0)\), which is 1, resulting in (1)^{2}.
04

Final result

The final simplified expression is (1)^{2}, which is equal to 1. Therefore, the limit of (\(\cos \frac{1}{x}\))^{2} as x approaches infinity is 1.

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Key Concepts

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

Cosine Function
The cosine function, represented as \( \cos(x) \), is one of the fundamental trigonometric functions. It is important in understanding wave patterns, circles, and oscillations. The cosine function varies between -1 and 1 for all real numbers.
  • The cosine function is periodic with a period of \( 2\pi \). This means its values repeat every \( 2\pi \) radians.
  • \( \cos(x) \) is even, which means \( \cos(-x) = \cos(x) \).
  • The maximum value of \( \cos(x) \) is 1, occurring at 0, \(2\pi\), etc.
  • The minimum value is -1, occurring at \(\pi\), \(3\pi\), etc.
It transitions smoothly as it goes through these maximum and minimum points, which is crucial in determining the behavior of the function as the input approaches certain critical values like zero.
Infinity Limit
A limit as a variable approaches infinity helps us understand the behavior of a function as the variable grows very large. In calculus, when we say \( x \rightarrow \infty \), it indicates that \( x \) increases without bound.

  • For a fraction with a constant numerator and an increasing denominator, such as \( \frac{1}{x} \), the value approaches zero as \( x \rightarrow \infty \).
  • This concept is used often to simplify complex expressions, as many components may become negligible as \( x \) grows.
  • Infinity limits are crucial when considering the end behavior of functions, particularly in real-world applications such as physics and engineering.
When applied to trigonometric functions, understanding infinity limits allows us to piece together how these functions behave across their domains, ultimately making it easier to predict and calculate results in specific scenarios.
Trigonometric Limits
Trigonometric limits involve calculating the value a trigonometric function approaches as the input tends towards a particular point or infinity. Sometimes, these involve transformations, like \,\( \frac{1}{x} \,\), seen in our original exercise.

  • The behavior of trigonometric functions as their arguments approach 0 or infinity is central to many calculus problems.
  • Consider \( \cos \left( \frac{1}{x} \right) \) as \( x \rightarrow \infty \): \( \frac{1}{x} \rightarrow 0 \), leading \( \cos \left( \frac{1}{x} \right) \rightarrow \cos(0) \).
  • Utilizing trigonometric limits involves thoughtful substitution to simplify and accurately find limits.
  • Common trigonometric functions approached through limits need careful handling, especially as values can rapidly oscillate.
Understanding these trigonometric limits helps formulate an approach for solving more intricate calculus problems, such as derivatives and integrals involving trigonometric identities.

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Most popular questions from this chapter

You have seen that for all real \(x\).$$\lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n}=e^{x}$$,However, the rate of convergence is different at different X. Verify that with \(n=100,(1+1 / n) ^ {n}\) is within \(1 \%\) of its \(\begin{array}{ll}\text { (1) } & \text { (i) } 4 \text { is twithin } 1 \% \text { the } \\ \text { the the the the the the the the parition the parition }\end{array}\) limit, while \((1+5 / n)^{n}\) is still about \(12 \%\) from its limit. Give comparable accuracy estimates at \(x=1\) and \(a x=5\) for \(n=1000\).

Use comparison test (11.7.2) to determine whether the integral converges. $$\int_{0}^{\infty}\left(1+x^{5}\right)^{-1 / 6} d x$$

Let \(\Omega\) be the region bounded by the curve \(y=x^{-1 / 4}\) and the \(x\) -axis, \(0=x \leq 1\). (a) Sketch \(\Omega\). (b) Find the area of \(\Omega\). (c) Find the volume obtained by revolving \(\Omega\) about the \(x\) -axis. (d) Find the volume obtained by revolving \(\Omega\) about the \(y\) -axis.

Sketch the curve, specifying all vertical and horizontal asymptotes. $$y=\frac{\ln x}{x}$$

Find the indicated limit.$$\begin{aligned}&\lim _{n \rightarrow \infty} \frac{1+2+\cdots+n}{n^{2}}\\\ &\text { HINT: } 1+2+\cdots+n=\frac{n(n+1)}{2}\end{aligned}$$.
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