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

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?

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64  
If the limit of $f'/g'$ exists, then it is also the limit of $f/g$. Not the other way around. – user251257 Mar 23 at 21:11
6  
One often forgets there are hypotheses to check before applying L'Hospital. One of these is that the ratio of the derivatives must exist (or still be indeterminate). – Bernard Mar 23 at 21:13
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A condition on the use of L'Hôpital in this context is that the derivative of the denominator must be non-zero on $(N, \infty)$ for some $N$. – Brian Tung Mar 23 at 21:15
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This post might give you something to think about. – Hirshy Mar 23 at 21:23
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An excellent example for a Calculus Course. – dwarandae Mar 24 at 4:46
up vote 116 down vote accepted

Your only error -- and it's a common one -- is in a subtle misreading of L'Hopital's rule. What the rules says is 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.

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59  
Not just a common error, a VERY common error. And I am from now on going to use the OP's example in my next calc batch! – imranfat Mar 23 at 21:18
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What's even funnier is examples where you apply l'Hopital twice and get back where you started :-) – gnasher729 Mar 25 at 21:11

L'Hopital's rule only tells you that if the modified limit exists and has value $L$, then the original limit also exists and has value $L$. It doesn't tell you that the converse holds.

So, the fact that the modified limit doesn't exist gives you no information about the original limit. So, you need a different method.

Consider something more direct: can you compute $$ \lim_{x\to\infty}\frac{x}{x+\sin x}=\lim_{x\to\infty}\frac{1}{1+\frac{\sin x}{x}}? $$

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Or 1 - sin x / (x + sin x). – gnasher729 Mar 24 at 22:16
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or squeeze theorem using -1 <= sin x <= 1. – djechlin Mar 25 at 18:57

De L'Hôpital's rule states that: if $f$ and $g$ are functions that are differentiable on an open interval $I$ (except possibly at a point $x_0$ contained in $I$), if $$\lim_{x\to x_0}f(x)=\lim_{x\to x_0}g(x)=0 \;\mathrm{ or }\; \pm\infty,$$ if $g'(x)\ne 0$ for all $x$ in $I$ with $x \ne x_0$, and $\lim_{x\to x_0}\frac{f'(x)}{g'(x)}$ exists, then:

$$\lim_{x\to x_0}\frac{f(x)}{g(x)} = \lim_{x\to x_0}\frac{f'(x)}{g'(x)}\,.$$

The most classical "counter-example" is when functions are constant: $f(x)=c$ and $g(x)=1$. The derivative of $g(x)$ vanishes on any open interval, while $f/g = c$.

The factorization proposed by @Nick Peterson typically avoids to resort to the rule when it is not necessary (especially when the inderterminacy can be lifted easily). It looks magic, and as for every magic, it shall be used with parsimony (unless it unleashes terrible powers).

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others already said that l'Hopital requires existence of the limit of the ratio of the derivatives; However in addition, with a solid understanding of limit definition is still possible to prove solution applying De l'Hopital, but not to that function, think about this:

$$\lim_{x \to +\infty} \frac{x}{x+1} \leq \lim_{x \to +\infty} \frac{x}{x-\sin(x)} \leq \lim_{x \to +\infty} \frac{x}{x-1}$$ condensed considering also $-\infty$ with $$\lim_{x \to \infty} \frac{x}{x+sig(x)} \leq \lim_{x \to \infty} \frac{x}{x+\sin(x)} \leq \lim_{x \to \infty} \frac{x}{x-sig(x)}$$ where $$sig(x)=\left\{ \begin{matrix} 0 & x=0\\ \frac{|x|}x & x\ne 0 \end{matrix} \right.$$

prove the above while apply l'Hopital to

$$\lim_{x \to \infty} \frac{x}{x\pm 1}$$

the squeezing inequities are true after a certain G, formally $\exists G / \forall x\in\Re,|x|>G : \frac{x}{x+sig(x)} \leq \frac{x}{x+\sin(x)} \leq \frac{x}{x-sig(x)}$

applying the limit definition to $x \over x+sin(x)$ the starting point M selecting all x>M has to be greater or equal than G (simply require $M\geq G$), in this case M=G is great enough to say that the limit is the same 1.

More formally (I actually didn't find an online pointable suitable formal definition of $\lim_{x\to\infty}$, so I'm making it up)

$$\lim_{x \to \infty} f(x) = r\in \{\Re, -\infty, +\infty, NaN\} / \\ \exists r \in \Re : \forall \epsilon \in \Re, \epsilon>0: \exists M \in \Re : \forall x \in \Re, |x| > M : |f(x)-r|<\epsilon \\ \lor r=\infty, omissis \\ \lor r=+\infty, omissis \\ \lor r=-\infty, omissis \\ \lor r=NaN, omissis. $$ (r as abbreviation of response, NaN (not a number) is when the limit doesn't exists and $\lor$ is in this case a shortcut or).

think of names

$f(x)=\frac{x}{x+\sin(x)}$

$g(x)=\frac{x}{x \pm 1}$, and when the definition of limit is used with g(x) the lower bound M is called G

from the evident property $\exists G' \in \Re^+ | \forall x \in \Re, |x|>G' : x-1 \leq x+\sin(x) \leq x+1$

$\Rightarrow \exists G \in \Re^+ | \forall x \in \Re, |x|>G : \frac{x}{x+sig(x)} \leq \frac{x}{x+\sin(x)} \leq \frac{x}{x-sig(x)}$

$$\lim_{x \to \infty} \frac{x}{x\pm 1} \underleftarrow{=(?H)= \lim_{x \to +\infty} \frac{\frac{d}{dx} x}{\frac{d}{dx}(x \pm 1)} = \lim_{x \to +\infty} \frac{1}{1 \pm 0}=1}$$ the existence of this limits (they are two, due to $\pm$) ensures that

$\forall \epsilon \in \Re, \epsilon>0: \exists G \in \Re : \forall x \in \Re, |x| > G : |g(x)-r|<\epsilon$

Choosing $M \geq G$ ($M$ is the lower bound in the definition of limit for $f(x)$) $$ \Rightarrow \lim_{x \to \infty} f(x)=1$$

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There is another useful rule, which I don't seem to have seen written down explicitly:

Let f, g, r, and s be functions where the g -> inf, r and s are bounded. Then taking the limit of f / g or the limit of (f + r) / (g + s) gives the same result.

Applied here, since sin (x) is bounded, the limit is the same as the limit of x / x.

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