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Chapter 3: Problem 63

Find the limit, if it exists. $$ \lim _{x \rightarrow \infty} \cos x $$

Short Answer

Expert verified
The limit does not exist.

Step by step solution

01

Understand the Function

The given function to find the limit of is \ \(\cos x \). This is a trigonometric function.
02

Recall the Behavior of the Cosine Function

Recognize that \ \(\cos x \) is bounded and oscillates between -1 and 1, regardless of the value of \ \(x \).
03

Evaluate the Limit as x Approaches Infinity

When \ \(x \) approaches infinity, \ \(\cos x \) does not settle to a single value because it continues to oscillate between -1 and 1. Thus, the limit does not exist.

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Key Concepts

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

Limit
The idea of a limit is fundamental in calculus. Essentially, we use limits to describe the behavior of a function as the input approaches a certain value.

The limit tells us what value a function approaches, but not necessarily reaches, as the input gets closer to some number.
In our exercise, the focus is on what happens to the value of \( \cos x \, \) as \( \x \, \) goes to infinity.

Here, we need to identify the pattern or behavior of the function \( \cos x \, \) and how it behaves when \( \x \, \) becomes very large.
Cosine Function
The cosine function, \( \cos x \, \), is one of the basic trigonometric functions.

Some important properties include:
  • It is periodic, meaning it repeats its values in regular intervals.
  • It oscillates, or moves back and forth, between -1 and 1.
  • The value of \( \cos x \, \) at any point is given by the horizontal distance from the origin on the unit circle.
For \( \cos x\, \), the period is \( \2\pi \, \), meaning the function completes one full cycle of its pattern every \( \2\pi \, \) units in the x direction.

The cosine function will keep repeating its pattern infinitely, no matter how large \( \x\, \) gets.
Oscillation
Oscillation refers to any function or signal that swings back and forth between two values.

For the case of \( \cos x \, \), the function oscillates between -1 and 1 continuously.
  • Due to its periodic nature, \( \cos x \, \) never settles at a single value.
  • Even as \( \x \, \) approaches infinity, the function does not converge to a particular number.
  • This continuous oscillation means we cannot pin down a limit for \( \cos x \, \) as \( \x \, \) goes to infinity.
The key takeaway is that a limit only exists if the function settles at a single value. Since \( \cos x \, \) keeps changing between -1 and 1, the limit as \( \x \, \) approaches infinity does not exist.

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Minimize \(Q=x^{3}+2 y^{3}\), where \(x\) and \(y\) are positive numbers, such that \(x+y=1\).

Then graph the tangent line to the graph at the point \((-0.8,0.384)\). $$ x^{3}=y^{2}(2-x) $$

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