Let f be a differentiable function. Prove that if \(\lim_{x\to\infty} (f(x)+f'(x))=1\), then \(\lim_{x\to\infty} f(x)=1\).

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Let f be a differentiable function. Prove that if \(\lim_{x\to\infty} (f(x)+f'(x))=1\), then \(\lim_{x\to\infty} f(x)=1\).

Find all real-valued continuous function f on the reals such that f(x)=f(cos x) for every real number x.

Find all real numbers \(\lambda\) and the corresponding functions \(f\) such that the equation