Asymptotic Equality: Can I Conclude a-c~b-d?

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In summary, "asymptotic equality" is a term used to describe the relationship between two functions as one approaches infinity. It means that the growth rate of one function is equal to or less than the growth rate of the other function. It differs from regular equality in that it compares the growth rates of functions rather than their values. Asymptotic equality can be used to compare any types of functions as long as they have a common behavior at infinity. It can be used in scientific research to analyze and compare different functions, make predictions, and identify patterns in data sets.
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If a,...,d are functions and a~b, c~d, can I conclude that a-c~b-d?
 
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No. Example a=b=xn, c=xn-xn-1, d=xn-xn-2

Fix n>1 and x->inf.
 
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No, you cannot conclude that a-c~b-d based on the given information. Asymptotic equality means that two functions have the same asymptotic behavior, but it does not necessarily mean that all operations between them will also have the same asymptotic behavior. In this case, we know that a~b and c~d, but we do not have any information about the asymptotic behavior of a-c or b-d. Therefore, we cannot make a conclusion about the asymptotic equality of a-c and b-d.
 

FAQ: Asymptotic Equality: Can I Conclude a-c~b-d?

1. What does "asymptotic equality" mean?

"Asymptotic equality" refers to the relationship between two functions as one approaches infinity. It means that the growth rate of one function is equal to or less than the growth rate of the other function.

2. How is asymptotic equality different from regular equality?

In regular equality, two values are exactly the same. In asymptotic equality, the values may be different, but their growth rates are the same as one approaches infinity.

3. Can asymptotic equality be used to compare different types of functions?

Yes, asymptotic equality can be used to compare any two functions as long as they have a common behavior as they approach infinity. This includes polynomial, exponential, and logarithmic functions.

4. What does it mean to conclude "a-c~b-d" using asymptotic equality?

This means that as a and b approach infinity, and c and d approach infinity, the growth rates of the functions a and b are equal to or less than the growth rates of the functions c and d. In other words, the growth rates of a and b are asymptotically equal to the growth rates of c and d.

5. How can I use asymptotic equality in my scientific research?

Asymptotic equality can be used to compare and analyze the behavior of different functions, which can be helpful in various fields of science such as physics, biology, and computer science. It can also be used to make predictions and identify patterns in data sets.

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