Limit as x tends towards +infinity for cos(1/x) to be 1

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In summary, the conversation is about finding the limit as x tends towards positive infinity for cos(1/x). The speaker has correctly calculated the limit to be 1, but the answer in the book lists the answer as (0,3) and (3,+infinity). The speaker questions if this is a misprint or if they have made a mistake. The conversation ends with the conclusion that the answer is most likely a misprint.
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grizz45
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I have worked out the limit as x tends towards +infinity for cos(1/x) to be 1, as cos (1/infinity) would be cos(0) which is 1. However the answer in the book with the question says that the answer should be (0,3) and (3,+infinty)! Is this a misprint or have i gine drastically wrong?
 
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grizz45 said:
I have worked out the limit as x tends towards +infinity for cos(1/x) to be 1, as cos (1/infinity) would be cos(0) which is 1. However the answer in the book with the question says that the answer should be (0,3) and (3,+infinty)! Is this a misprint or have i gine drastically wrong?

Of course
[tex]\lim_{x \to \infty} \cos(1/x)[/tex]
doesn't tend toward (0,3) and (3,infty) (how would that make any sense?). That is probably the answer to a different problem (have you checked the problem number and chapter number is correct?). Your idea is correct except for the fact that 1/infinity makes no sense, but since cos is continuous and defined in 0 you have:
[tex]\lim_{x \to \infty} 1/x = 0 \Rightarrow \lim_{x \to \infty} \cos(1/x) = \cos(0) = 1[/tex]
so your answer is correct.
 
  • #3


i have checked the problem number and everything...i think there is a massive misprint as i have done some other questions from the same section and the answers all seem to be wrong as there are of the same format as the answer in the first post!
 

Related to Limit as x tends towards +infinity for cos(1/x) to be 1

1. What is the definition of a limit as x tends towards +infinity?

The limit as x tends towards +infinity is a mathematical concept that represents the value that a function approaches as its input (x) gets infinitely large. It is denoted by the notation "lim x→+∞ f(x)" and is used to describe the behavior of a function at the end of its domain.

2. How is the limit as x tends towards +infinity calculated?

The limit as x tends towards +infinity can be calculated by evaluating the function at increasingly large values of x. If the values of the function approach a specific number, then that number is the limit. If the values do not approach a specific number, the limit does not exist.

3. What is the significance of the limit as x tends towards +infinity for cos(1/x) to be 1?

The significance of the limit as x tends towards +infinity for cos(1/x) to be 1 is that it represents the behavior of the cosine function as its input gets infinitely large. This limit indicates that the function approaches the value of 1 as x gets larger and larger, regardless of the exact value of x.

4. What does the value of 1 represent in the limit as x tends towards +infinity for cos(1/x) to be 1?

The value of 1 in the limit as x tends towards +infinity for cos(1/x) to be 1 represents the horizontal asymptote of the cosine function. This means that the function will never cross the line y=1 as x gets infinitely large.

5. Can the limit as x tends towards +infinity for cos(1/x) to be 1 be proven mathematically?

Yes, the limit as x tends towards +infinity for cos(1/x) to be 1 can be proven mathematically using the epsilon-delta definition of a limit. This involves showing that for any small positive number (epsilon), there exists a large positive number (delta) such that the distance between cos(1/x) and 1 is less than epsilon whenever x is larger than delta.

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