What is infinity to the power of zero?

In summary, the result of raising infinity to the power of zero is undefined. This is because infinity is not a legitimate number in the real number system, and while there are extended number systems where infinity is defined, the usual arithmetic operations do not apply. Instead, we use limits to express such results, but these expressions are considered indeterminate until a specific question defines them.
  • #1
Pyroadept
89
0

Homework Statement


Hi everyone,

I'm just wondering if someone could please clarify for me what infinity to the power of zero is? I seem to be finding conflicting opinions about this online. Is it '1' or 'not defined'?

Thanks!


Homework Equations





The Attempt at a Solution

 
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  • #2
Infinity is not a legitimate number with which you could ask what the result of raising it to the zero power is.
 
  • #3
Thanks!
 
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More specifically, "infinity" is not a member of the real number system on whicy our standard operations are defined. There do exist "extended" number systems in which "infinity" is defined but then the usual arithmetic operations to not apply.
 
  • #5
Pyroadept said:

Homework Statement


Hi everyone,

I'm just wondering if someone could please clarify for me what infinity to the power of zero is? I seem to be finding conflicting opinions about this online. Is it '1' or 'not defined'?

Thanks!

Homework Equations


The Attempt at a Solution

It's undefined. Also we never say what infinite to the power of zero is, we instead express such results by the use of limits:

[tex]\lim_{x\to \infty}x^{1/x}[/tex] is such an expression that would be of the form [itex]\infty ^0[/itex] and in this case it's equal to 1, but there are many other cases where it's not, such as [tex]\lim_{x\to \infty}\left(x^x\right)^{1/x}=\infty[/tex] or [tex]\lim_{x\to \infty} x^{1/\ln(x)}[/tex] which is a special one that equals the irrational number [itex]e\approx 2.718[/itex]

This is why expressions of this form are called indeterminate. They're undefined, until you find the specific question that defines them. It's different from the sense of 1/0 being undefined since that one is undefinable since it would create inconsistencies in our maths.
 

FAQ: What is infinity to the power of zero?

What is infinity to the power of zero?

Infinity to the power of zero is a mathematical expression that represents the limit of a function as its input approaches zero. It is often written as 1/0, but this is undefined in most cases.

Is infinity to the power of zero equal to zero?

No, infinity to the power of zero is not equal to zero. In fact, it is undefined and cannot be assigned a numerical value.

Is infinity to the power of zero equal to infinity?

No, infinity to the power of zero is also not equal to infinity. It is important to remember that infinity is not a number, it is a concept that represents something without an end. Therefore, it cannot be used in mathematical operations like raising to a power.

Can infinity to the power of zero be calculated?

No, as mentioned before, infinity to the power of zero is undefined and cannot be calculated. It is important to be careful when using infinity in mathematical expressions, as it can lead to incorrect or undefined results.

What is the value of infinity to the power of zero?

As infinity to the power of zero is undefined, it does not have a specific value. It is important to keep in mind that infinity is not a number and cannot be used in the same way as numerical values in mathematical operations.

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