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The following limit evaluates to 1 $$\lim_{x\to\infty}\frac{x + \sin x}{x} = 1 + \lim_{x\to\infty}\frac{\sin x}{x} = 1$$
But when I use L'Hopital's rule, it doesn't: $$\lim_{x\to\infty}\frac{x+\sin x}{x} = \lim_{x\to\infty}\frac{1+\cos x}{1} = 1+\lim_{x\to\infty}\cos x$$
Why doesn't L'Hopital's rule work here? What conditions does it not satisfy?
L'Hopital's Rule says that if the limit $\lim_{x\to\infty}\frac{f'(x)}{g'(x)}$ exists, then $\lim_{x\to\infty}\frac{f(x)}{g(x)}=\lim_{x\to\infty}\frac{f'(x)}{g'(x)}$. In your case, the limit $\lim_{x\to\infty}\frac{f'(x)}{g'(x)}$ does not exist. So, there is no contradiction here.
