Limit as X-> infinity of cos(x)

In summary, the limit as x approaches infinity of cos(x) does not exist because the cosine function oscillates between -1 and 1 infinitely as x increases without bound. It cannot be calculated or equal to 0, but it can be negative due to the cosine function's behavior in the second and third quadrants. Furthermore, it does not have any real solutions since the limit does not exist.
  • #1
Quantum_Grid
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limit as X--> infinity of cos(x)

Homework Statement


Find the limit.
[tex]\stackrel{lim}{x\rightarrow\infty}cos(x)[/tex]


The Attempt at a Solution


As best as I can tell by putting larger and larger numbers in my calculator, there is no limit, as cos(x) just oscillates between (-1,1). So is the limit undefined, or is it 1?
 
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  • #2


Your first thought is correct. It is undefined.
 

1. What is the limit as x approaches infinity of cos(x)?

The limit as x approaches infinity of cos(x) does not exist. This is because the cosine function oscillates between -1 and 1 infinitely as x increases without bound, and therefore does not approach a single value.

2. How do you calculate the limit as x goes to infinity of cos(x)?

The limit as x goes to infinity of cos(x) cannot be calculated, as it does not exist. However, if the limit were to exist, it could be calculated by evaluating the function at increasing values of x to see if it approaches a single value.

3. Is the limit as x goes to infinity of cos(x) equal to 0?

No, the limit as x goes to infinity of cos(x) is not equal to 0. As mentioned before, the cosine function oscillates between -1 and 1 infinitely, so it does not approach a single value and therefore the limit does not exist.

4. Can the limit as x goes to infinity of cos(x) be negative?

Yes, the limit as x goes to infinity of cos(x) can be negative. This is because the cosine function is negative in the second and third quadrants of the unit circle, which are repeated infinitely as x increases without bound.

5. Does the limit as x goes to infinity of cos(x) have any real solutions?

No, the limit as x goes to infinity of cos(x) does not have any real solutions. This is because the limit does not exist, so it cannot be equal to any single real value.

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