Does $\exp(-x) \sin(x)$ have infinite oscillations AND an infinite number of real roots

Question

A Course of Pure Mathematics by G. H. Hardy states that a function oscillates if it does NOT tend to a limit or to $± \infty$. I find this definition difficult to square with $e^{-x}\sin(x)$. It has limit $0$ as $x$ tends to positive infinity and also an infinite number of roots, located at integer multiples of $π$ and as such it must have an infinite number of turning points and by extension must oscillate between positive and negative values indefinitely. I welcome comments which clarify this apparent 'contradiction'.

Answer 1

This is Hardy's definition copied from Project Gutenberg's A Course of Pure Mathematics.

62. Oscillating Functions. $\text{Definition.}$ When $ϕ(n)$ does not tend to a limit, nor to $+ ∞$, nor to $−∞$, as $n$ tends to $∞$, we say that $ϕ(n)$ oscillates as $n$ tends to $∞$.

According to this definition, the function $\,f(x)=e^{-x} \sin(x)\,$ does not oscillate when $\,x \to \infty\,$, since the limit at $\,+\infty\,$ exists, and is in fact $\,0\,$.

A related concept is that of oscillation of a function, and the oscillation of this $\,f(x)\,$ at $\,+\infty\,$ is $\,0\,$. Hardy's definition is equivalent to saying that a function oscillates at $\,+\infty\,$ iff the oscillation at $\,+\infty\,$ is finite and strictly greater than zero.

It is true that $\,f(x)\,$ changes sign infinitely many times, and does so in every neighborhood of $\,+\infty\,$, but it "flatlines" and has the horizontal axis as an asymptote. Such functions are sometimes caled damped oscillations, though it would be less common to call them "oscillating functions" generically (nor does the OP quote any specific source that uses an alternative definition under which $\,f(x)\,$ would be called an "oscillating function").