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

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The discussion clarifies that if functions a, b, c, and d are asymptotically equal (a~b and c~d), one cannot conclude that a-c is asymptotically equal to b-d. An example provided is with a = b = x^n, c = x^n - x^(n-1), and d = x^n - x^(n-2), where n > 1 and x approaches infinity. This demonstrates that the asymptotic relationships do not hold under subtraction.

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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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