The M1M1 treasure hunt, week 2

Alessio sat in a South Kensington cafe enjoying the last of the summer sun early one October morning. He gazed out at the cheerful, youthful faces, so keen to start their university studies not far away. Was it so very long ago, he mused, since he himself had been one of them? He recalled clearly the excitement, the challenge, the allure of the brave new mathematical world before him. They had called him "Aless" in those days. His eyes misted over, and then gently closed, a serene smile upon his face, as his mind drifted back to that carefree time....

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"Oh my ears and whiskers!" exclaimed a voice nearby. "I'll be late for my 9 O'Clock lecture." Startled, Aless jumped up, just in time to see a white rabbit vanishing into a dark tunnel underneath Brompton Rd. Bemused, Aless followed the curious creature, down into the bowels of the earth. After a long walk, they emerged into a strange building, around which various mysterious beings bustled busily.
"How interesting they all look!" thought Aless. "Won't it be exciting to meet more of them in the weeks to come."
The white rabbit darted into a packed theatre and quietly sat down in a newly refurbished seat. Aless did the same. At the front of the hall, someone was giving a lecture about polynomials. He seemed excited about the fact that an N^th order polynomial always has N roots.
"Well of course it does," thought Aless crossly. "That's how many x's there are. But I wonder what happens when N gets very big? Does an infinite polynomial have infinitely many roots? Suppose I make up a function for which every integer is a root. Say

f(x)=(M+x)(M-1+x)...(1+x)(x)(1-x)...(M-1-x)(M-x)

Soon I'll let M → ∞, but maybe I should divide by something first to stop the function getting too big. I know, I'll divide by the gradient at zero, and define
g(x)=f(x)/f'(0)."

What is the slope of the curve y = f(x) at the origin, i.e. what is f '(0), evaluated (say) for M=5?

f '(0) = ?

"Now I wonder by how much the function g(x) changes if I increase x by 1," thought Aless. "I'll calculate the ratio g(x+1)/g(x) carefully, always assuming both values aren't zero, of course. Then I'll let M → ∞, and add one for luck."

Lim_{M → ∞}[g(x+1) / g(x)] + 1 = ?

"Curiouser and curiouser!" exclaimed Aless. "When M is very big that ratio hardly changes at all. That must surely mean that if this polynomial has a limit, then that limiting function is periodic, with period 2. As it was me who found it, I'll call the limit the 'Aless Function', A_le(x):"

A_le(x)=Lim_M→∞[g(x)]

"So what do I know? My function A_le(x) is 2-periodic, is zero for every whole number and A_le'(0)=1." Aless experimented a bit with a calculator, and discovered also that A_le(0.5) ≈ 0.32.

After a bit of thought, Aless made an intelligent guess as to what the function A_le(x) might be equal to:

A_le(x) = ?