"How would you best describe the rate of growth of the function f(x)=cxr^x? Consider the function f(x)=cxrx, where both r and c are constants and we have cases: (a) r<1, (b) r>1. Regarding terminology, how would you best describe the asymptotic growth of f(x) in cases (a) and (b)? Though you could state that exponential growth and decay will dominate the asymptotics of the function, strictly speaking it would be incorrect to call (a) exponential growth and (b) exponential decay, right?"

Kenna Stanton

Kenna Stanton

Answered question

2022-11-02

How would you best describe the rate of growth of the function f ( x ) = c x r x ?
Consider the function f ( x ) = c x r x , where both r and c are constants and we have cases: (a) r<1, (b) r>1. Regarding terminology, how would you best describe the asymptotic growth of f(x) in cases (a) and (b)? Though you could state that exponential growth and decay will dominate the asymptotics of the function, strictly speaking it would be incorrect to call (a) exponential growth and (b) exponential decay, right?

Answer & Explanation

Faith Wise

Faith Wise

Beginner2022-11-03Added 17 answers

Exponential growth and exponential decay when x + are often defined by the fact that
lim x + log f ( x ) x
exists and is not zero. In this context, every function f : x c x a ( log x ) b r x with c>0 and r>0, r 1, has exponential growth or exponential decay when x + .

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