Eta BEta

Calculus Level 3

Let \(f(x) =k\cos x\). If the sum of coefficients of \(\cos x\) in the expansion of \(\left(1+f(x)\right)^n\) is \(\eta\) and the sum of coefficients of \(\sin x\) in the expansion of \(\left(1+\dfrac {df(x)}{dx}\right)^n\) is \(\beta\), find

\[\large \lim_{n\to\infty} \frac{|\eta\beta|}{\left(|\eta\beta|^{1/n}+1\right)^n}.\]


This is an original problem, and resides in the set Sravanth's Challenges.
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