Does the Limit of x^4 * 0.99^x as x Approaches Infinity Equal Zero?

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In summary, the conversation is about finding the limit of a function, F(x) = a^x, as x approaches positive infinity. The value of a is between 0 and 1. The original question was whether the limit of this function is equal to 0. The conversation discusses the importance of defining the function, as well as the fact that 0.99x goes to 0 faster than x^4 goes to infinity. In the end, it is determined that the limit is indeed 0.
  • #1
JPC
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hey

if let's say : F(x) = 0.5^x , 0 < 0.5 < 1

is lim(+infinte) f = 0 ?

.............

this is for one of my math questions :

lim (+infinite) x^4 * 0.99^x = ?
 
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  • #2
Impossible to say, since you haven't said what f is, only F.

And yes, we DO know that 0.5 lies between 0 and 1.

What is meant by infinite indice is beyond me.
 
  • #3
WE have the sequence 1/2, 1/4, 1/8...,obviously the nth term gets as close to zero as we like, and so 0 is the least upper bound and since it is never less than zero, the limit is 0.
 
  • #4
ok then

lim (+infinite) x^4 * 0.99^x = 0

thanks
i just had a little dougthBTW : arildno ;
- for the F and f problem
> i just made a caps mistake
- for the 0 < 0.5 < 1
> it was because the F(x) = 0.5^x was an example for any function F(x)= a^x, where a is a real number that respects (0 < a < 1)
- for "infinite indice"
> it was just to make short
 
  • #5
robert Ihnot said:
WE have the sequence 1/2, 1/4, 1/8...,obviously the nth term gets as close to zero as we like, and so 0 is the least upper bound and since it is never less than zero, the limit is 0.

No, 0 is the greatest lower bound. 1/2 is the least upper bound.
 
  • #6
JPC said:
ok then

lim (+infinite) x^4 * 0.99^x = 0

thanks
i just had a little dougth


BTW : arildno ;
- for the F and f problem
> i just made a caps mistake
- for the 0 < 0.5 < 1
> it was because the F(x) = 0.5^x was an example for any function F(x)= a^x, where a is a real number that respects (0 < a < 1)
- for "infinite indice"
> it was just to make short

If you mean [itex]\lim_{x\right arrow +\infnty} x^4 * 0.99^x[/itex] then it is true that 0.99x goes to 0 but it does NOT follow from that alone that the whole thing goes to 0 because x^4 goes to infinity.

It happens that the limit of x40.99x is 0 because 0.99x goes to 0 faster than x4 goes to infinity0- but that has to be shown.

If, as your use of "x" rather than "n" indicates, you intended this to be a continuous limit, then x is not an "index" at all. (There is no such word as "indice" in English. "Indices" is the plural of "index".
 

FAQ: Does the Limit of x^4 * 0.99^x as x Approaches Infinity Equal Zero?

What is the "Infinite Indice Problem"?

The "Infinite Indice Problem" is a mathematical concept that deals with the number of possible combinations that can be created from an infinite set of elements. It is also known as the "Hilbert's Infinite Hotel Paradox" and was first introduced by mathematician David Hilbert in the early 1900s.

How does the "Infinite Indice Problem" work?

The "Infinite Indice Problem" works by imagining a hotel with an infinite number of rooms, each occupied by a guest. If a new guest arrives, the hotel manager can accommodate them by simply shifting all the current guests to the next room number, making room for the new guest in room number one. This paradox highlights the concept of infinity and its counterintuitive properties.

What are some real-world applications of the "Infinite Indice Problem"?

The "Infinite Indice Problem" has applications in various fields such as mathematics, physics, and computer science. In mathematics, it is used to solve problems related to infinite sets and cardinality. In physics, it is used to understand the concept of infinity in space and time. In computer science, it is used to develop algorithms and data structures for handling infinite data sets.

What are some criticisms of the "Infinite Indice Problem"?

One criticism of the "Infinite Indice Problem" is that it is a purely theoretical concept and does not have any practical relevance. Some argue that infinity is an abstract concept and cannot exist in reality. Others criticize the paradox for relying on the assumption that an infinite number of guests can be accommodated in a finite space, which is not physically possible.

Are there any solutions to the "Infinite Indice Problem"?

There is no definitive solution to the "Infinite Indice Problem" as it is a paradox that challenges our understanding of infinity. However, mathematicians and philosophers have proposed various theories and arguments to address the paradox, such as the concept of potential infinity and the use of mathematical logic and set theory. Ultimately, the "Infinite Indice Problem" remains an ongoing debate and continues to fascinate and puzzle scientists and philosophers alike.

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