- #1
Lisa91
- 29
- 0
[tex]\lim_{n\to\infty} \left (\lim_{k\to\infty} \cos (\left| n! \pi x\right|) ^{2k} \right) = \begin{cases} 1&x \in \mathbb{Q} \\0& x \not\in \mathbb{Q}\end{cases} [/tex]. How to prove it?
Last edited by a moderator:
Lisa91 said:[tex]\lim_{n\to\infty} \left (\lim_{k\to\infty} \cos (\left| n! \pi x\right|) ^{2k} \right) = \left\{\begin{array}{l} 1 x \in \mathbb{Q}\\1 x \not\in \mathbb{Q}\end{array}\right. [/tex]. How to prove it?
Lisa91 said:[tex]\lim_{n\to\infty} \left (\lim_{k\to\infty} \cos (\left| n! \pi x\right|) ^{2k} \right) = \left\{\begin{array}{l} 1 \in \mathbb{Q}\\ 0 \not\in \mathbb{Q}\end{array}\right. [/tex]. How to prove it?
Lisa91 said:[tex]\lim_{n\to\infty} \left (\lim_{k\to\infty} \cos (\left| n! \pi x\right|) ^{2k} \right) = \begin{cases} 1&x \in \mathbb{Q} \\0& x \not\in \mathbb{Q}\end{cases} [/tex]
No, this '2k' has to be in the place I wrote.
The limit of cosine squared is equal to 1 as x approaches infinity.
Proving the limit of cosine squared is important because it helps us understand the behavior of this trigonometric function and its relationship with other mathematical concepts.
The main difference is in the approach used to prove the limit. For rational numbers, we can use algebraic manipulation and properties of limits to prove the limit of cosine squared. However, for non-rational numbers, we need to use more advanced techniques such as the Squeeze Theorem or the ε-δ definition of a limit.
Yes, the limit of cosine squared can be proven for all real numbers. This is because the cosine function is defined for all real numbers and its square will always be positive, hence approaching 1 as x approaches infinity.
Proving the limit of cosine squared for non-rational numbers is significant because it expands our understanding of limits and allows us to apply more advanced techniques in mathematical analysis. It also helps us to better understand the behavior of trigonometric functions and their relationship with other mathematical concepts.