Problem

Let $f(x)=e^x+\sin(2x)$ and $g(x)=e^x+\cos(2x)$ for $x\in(-\infty,\infty)$. Which of the following statements is true?

(a) $-\infty<\lim_{x\to\infty} g(x)/f(x)\le 1$.

(b) $1<\lim_{x\to\infty} g(x)/f(x)\le 2$.

(c) $2<\lim_{x\to\infty} g(x)/f(x)\le 3$.

(d) $3<\lim_{x\to\infty} g(x)/f(x)<\infty$.

(e) $\lim_{x\to\infty} g(x)/f(x)$ does not exist.