133
$\begingroup$

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

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?

$\endgroup$
16
  • 98
    $\begingroup$ If the limit of $f'/g'$ exists, then it is also the limit of $f/g$. Not the other way around. $\endgroup$
    – user251257
    Commented Mar 23, 2016 at 21:11
  • 12
    $\begingroup$ 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). $\endgroup$
    – Bernard
    Commented Mar 23, 2016 at 21:13
  • 4
    $\begingroup$ 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 $N$. $\endgroup$
    – Brian Tung
    Commented Mar 23, 2016 at 21:15
  • 2
    $\begingroup$ This post might give you something to think about. $\endgroup$
    – Hirshy
    Commented Mar 23, 2016 at 21:23
  • 6
    $\begingroup$ An excellent example for a Calculus Course. $\endgroup$
    – dwarandae
    Commented Mar 24, 2016 at 4:46

5 Answers 5

168
$\begingroup$

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.

$\endgroup$
3
  • 81
    $\begingroup$ 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! $\endgroup$
    – imranfat
    Commented Mar 23, 2016 at 21:18
  • 14
    $\begingroup$ What's even funnier is examples where you apply l'Hopital twice and get back where you started :-) $\endgroup$
    – gnasher729
    Commented Mar 25, 2016 at 21:11
  • $\begingroup$ @gnasher729 Can you give us an example, I can't think of any $\endgroup$
    – Some Guy
    Commented Mar 4, 2021 at 6:53
45
$\begingroup$

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 $$ \lim_{x\to\infty}\frac{x}{x+\sin x}=\lim_{x\to\infty}\frac{1}{1+\frac{\sin x}{x}}? $$

$\endgroup$
2
  • $\begingroup$ Or 1 - sin x / (x + sin x). $\endgroup$
    – gnasher729
    Commented Mar 24, 2016 at 22:16
  • 2
    $\begingroup$ or squeeze theorem using -1 <= sin x <= 1. $\endgroup$
    – djechlin
    Commented Mar 25, 2016 at 18:57
12
$\begingroup$

Because you don't need it! And because one of the hypotheses (under which this technique classically applies) is not verified in this case.

Here, in layman terms, the ratio of functions, and the ratio of derivatives as well, does not have a clear enough limit. The de l'Hospital rule is more than often misused: people believe that if one cannot compute the limit of a ratio of functions, it is easier to compute the limit of the ratio of their derivatives. Possibly because derivatives sometimes look simpler, as for polynomials. Or similar, as for sines and cosines.

However, this is not true in general. Derivatives are rarely more continuous than their original functions (my fundamental anti-theorem of analysis).

Thus said, remember that the purpose of exercises is to train your mathematical skills, not to get the result. The teacher knows it already (hopefully). Using de L'Hôpital's rule is sometimes overkill, with which you don't learn what is going on with your functions. It is more efficient, and sounder, to try first simpler techniques, such as simplification, factorization of leading terms (here $x$), basic transformations (logarithms), etc.

Now let us go to the point.

De L'Hôpital's rule states that: if $f$ and $g$ are functions that are differentiable on some (small enough) 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 this overkill rule when it is not necessary (especially when the indeterminacy can be lifted easily).

De L'Hôpital's rule looks magic, and as for every magic, it shall be used wisely with parsimony (unless it unleashes terrible powers).

Here, tracks are legion. One very simple argument is that $x$ grows as... $x$, and $\sin(x)$ is bounded between $-1$ and $1$. So for a big enough $x$, $x-1\le x+\sin(x) \le x+1$, and you know that$ \frac{x}{x+1} $ and $\frac{x}{x-1}$ clearly tends to $1$ as $x\to \infty$.

$\endgroup$
9
$\begingroup$

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 $f(x)$) $$ \Rightarrow \lim_{x \to \infty} f(x)=1$$

$\endgroup$
8
$\begingroup$

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

Let $f, g, r$ and $s$ be functions such that $g\to\infty$ and $r, s$ are bounded.
Then the limit of $\dfrac{f}{g}$ and the limit of $\dfrac{f + r}{g + s}$ gives the same result.

Applied here, since $\sin x$ is bounded, the limit is the same as the limit of $\dfrac{x}{x}$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .