What is the limit of a circumference as the power approaches infinity?

  • MHB
  • Thread starter Julio1
  • Start date
  • Tags
    Limit
In summary, the conversation discusses the limit of a function $C_p$ as $p$ approaches infinity. It is believed that the limit is infinite, but the method for proving this is unknown. One way to show this is by using an equivalent expression involving the maximum of the absolute values of $x$ and $y$. However, it is questioned if the definition $\varepsilon-N$ can be used. It is then argued that the expression may not be true, as the limit of $|x|^p$ is 0 when $|x|<1$.
  • #1
Julio1
69
0
Show that $\displaystyle\lim_{p\to +\infty}C_p ((0,0); 1)=C_{\infty}((0,0);1).$

Hello, I think that this limit is infinite. At $C_p=\{(x,y)\in \mathbb{R}^2: |x|^p+|y|^p=1, \, p>1\}$, then is reasonably think. But how can show this?
 
Physics news on Phys.org
  • #2
An expression equivalent is $\displaystyle\lim_{p\to +\infty} \left|x \right|^p+\left|y \right|^p=\max\{\left|x \right|, \left|y \right|\}.$

Can use the definition $\varepsilon-N$?
 
  • #3
Julio said:
An expression equivalent is $\displaystyle\lim_{p\to +\infty} \left|x \right|^p+\left|y \right|^p=\max\{\left|x \right|, \left|y \right|\}.$
I don't think this is true. If $|x|<1$, then $\lim_{p\to\infty}|x|^p=0$.
 

FAQ: What is the limit of a circumference as the power approaches infinity?

What is the definition of the limit of a circumference?

The limit of a circumference is the value that a sequence of circumferences approaches as the number of sides of a regular polygon inscribed within the circumference increases infinitely. It represents the maximum possible circumference of a shape.

How is the limit of a circumference calculated?

The limit of a circumference is calculated by using the formula lim n→∞ (n * s), where n is the number of sides in the inscribed regular polygon and s is the length of each side. This formula is based on the fact that as the number of sides increases, the regular polygon more closely approximates a circle, thus giving a more accurate value for the limit of the circumference.

What is the significance of the limit of a circumference in geometry?

The limit of a circumference is significant in geometry because it allows us to determine the maximum possible circumference of a shape, even if it has an infinite number of sides. This concept is important in understanding the properties and limits of curves in mathematics.

Can the limit of a circumference be greater than the actual circumference of a shape?

No, the limit of a circumference cannot be greater than the actual circumference of a shape. The limit is the maximum possible circumference that a shape can have, and it can never exceed this value.

How does the limit of a circumference relate to the concept of infinity?

The limit of a circumference is closely related to the concept of infinity because it represents the maximum value that a circumference can approach as the number of sides increases infinitely. It is also used to define the concept of a circle, which is considered to have an infinite number of sides.

Similar threads

Back
Top