Why does L’Hopital’s rule fail in calculating $\lim_{x \to \infty} \frac{x}{x+\sin(x)}$?

$$\lim_{x \to \infty} \frac{x}{x+\sin(x)}$$

This is of the indeterminate form of type $\frac{\infty}{\infty}$, so we can apply l’Hopital’s rule:

$$\lim_{x\to\infty}\frac{x}{x+\sin(x)}=\lim_{x\to\infty}\frac{(x)’}{(x+\sin(x))’}=\lim_{x\to\infty}\frac{1}{1+\cos(x)}$$

This limit doesn’t exist, but the initial limit clearly approaches $1$. Where am I wrong?

Answer

Your only error — and it’s a common one — is in a subtle misreading of L’Hopital’s rule. What the rule says is that IF the limit of $f’$ over $g’$ exists then the limit of $f$ over $g$ also exists and the two limits are the same. It doesn’t say anything if the limit of $f’$ over $g’$ doesn’t exist.

Attribution
Source : Link , Question Author : Andrew Fount , Answer Author : Joe

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