How would you best describe the rate of growth of the function $f(x) = cxr^x$?

Question

Consider the function $f(x) = cxr^x$, 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 1

Exponential growth and exponential decay when $x\to+\infty$ are often defined by the fact that $$ \lim_{x\to+\infty}\frac{\log f(x)}x $$ exists and is not zero. In this context, every function $f:x\mapsto cx^a(\log x)^br^x$ with $c\gt0$ and $r\gt0$, $r\ne1$, has exponential growth or exponential decay when $x\to+\infty$.