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Consider $$\lim_{x\to\infty}\frac{f(x)}{g(x)} + \lim_{x\to\infty}\frac{h(x)}{i(x)}$$ I was told that I cannot apply L'Hôpital's rule to each individual limit and then join the limits as $$\lim_{x\to\infty}\frac{f(x)}{g(x)} +\frac{h(x)}{i(x)}$$ Why is this incorrect?

snorepion
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marzano
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4 Answers4

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I was told that I cannot apply L'Hôpital's rule to each individual limit and then join the limits as ...

This has nothing to do with the L'Hôpital's rule itself.

The rule that you cannot use is:

$\lim\limits_{x\to...}f(x)+\lim\limits_{x\to...}g(x)=\lim\limits_{x\to...}(f(x)+g(x))$

And you can see from JDMan4444's answer that there are situations where this rule does not work.

However, if you are sure that $\lim\limits_{x\to...}f(x)$ and $\lim\limits_{x\to...}g(x)$ exist (and are not $\pm\infty$), you can apply that rule. (And of course you can apply L'Hôpital's rule to the sum in this case.)

This is important because in some cases it is possible to prove that both limits exist but calculating the limits directly is very difficult or even impossible. In such cases calculating the limit of the sum may be easier.

Martin Rosenau
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Consider the following example:

$$ \lim_{x\rightarrow\infty}\frac{x^2}{x}+\lim_{x\rightarrow\infty}\frac{-x^2}{x} = \infty - \infty \quad \text{(which is undefined)} $$ $$ \lim_{x\rightarrow\infty}\frac{x^2}{x}+\frac{-x^2}{x} = 0 $$

Tanner Swett
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JDMan4444
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  • Ypu example is fine and correct but note that the main doubt was in the application of l'HR and is related to this other OP. – user Nov 29 '18 at 07:44
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    Your second line "$\lim_{x\rightarrow\infty}\frac{x^2}{x}+\frac{-x^2}{x} = 0$" is wrong. Addition has lower precedence than the limit operator, by convention. – user21820 Nov 29 '18 at 13:51
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    @user21820 indeed; round brackets are needed around both fractions together, otherwise it's +∞ because the limit applies only to the left fraction, the right remains -x^2/x. But that was obvious from the context. More importantly, this applies to the original question as much as here, so comment should be there. – user3445853 Nov 29 '18 at 15:17
  • @user21820 It could be convention but since the expression $\frac{x^2}{x}+\frac{-x^2}{x}$ is identically equal to zero for all $x\neq 0$ I think that it is a noce assumption read that as $$\lim_{x\rightarrow\infty} \left(\frac{x^2}{x}+\frac{-x^2}{x}\right) =\lim_{x\rightarrow\infty} 0= 0$$ – user Nov 29 '18 at 15:29
  • @user3445853: No. Teachers making errors are one big reason why students don't learn. I've nothing more to say here. – user21820 Nov 29 '18 at 16:47
  • @user21820 This isn't really an error, just notation that I, and many other mathematicians, use when context makes it clear what we are calculating. As stated by user3445853 even the OP used that notation. Your hostility seems a little unjustified because of it. – JDMan4444 Nov 29 '18 at 17:18
  • @JDMan4444: I'm not hostile; I'm stating a fact backed by years of teaching experience. Moreover, neither of you have gotten the point that students of course use wrong notation, largely because their teachers don't teach them properly. Have you never seen students who write "If $n$ is true, then ..." in a purported induction proof? – user21820 Nov 29 '18 at 18:02
  • @user21820 No, I have not seen that particular error in an induction proof. I also understand the point you were trying to make, but I dispute the idea that students have trouble learning math because of notational "sloppiness" that, in my opinion, isn't even that sloppy. I take it you also have objections with the equality $v'(t)=dv/dt$, since one is a number and the other is a function? While that distinction is important at some level, it is really a splitting of hairs that can get in the way of learning as I see it. – JDMan4444 Nov 29 '18 at 18:26
  • @user21820 Also of relevance to this conversation is the following post: https://math.stackexchange.com/questions/264610/why-is-abuse-of-notation-tolerated – JDMan4444 Nov 29 '18 at 18:49
  • @JDMan4444: I consider a notation sloppy if it is not applied consistently. In my opinion, one should explain to students precisely how to interpret each syntax, in an unambiguous manner. Otherwise we face all sorts of issues with things like $\sin(x^2+1)^r$ and $\prod_{i=1}^m i·\sum_{j=1}^n j$ and so on. There are ways to deal systematically and unambiguously with various notational devices (contrary to the implication of the answer at the link you gave) that are useful for mathematics. As for $v'(t) = dv/dt$, both sides are values, not functions... – user21820 Nov 29 '18 at 18:58
  • @user21820 An interesting take on the situation. As for the final point, $dv/dt$ is certainly not a value. It doesn't even have an argument to evaluate at, whereas $v'(t)$ is quite explicitly evaluated at $t$. – JDMan4444 Nov 29 '18 at 20:01
  • @JDMan4444: Short answer: you definitely recognize the use of phrases like "$dv/dt < 0$ when $t > 2$", and obviously this only makes sense if "$dv/dt$" is a real in the context where "$t > 2$", otherwise comparing to $0$ is meaningless. To avoid lengthening the comments, let's continue in chat. – user21820 Nov 30 '18 at 07:08
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The following identity

$$\lim_{x\to \infty}\left(\frac{f(x)}{g(x)} +\frac{h(x)}{i(x)}\right)=\lim_{x\to \infty}\frac{f(x)}{g(x)} + \lim_{x\to \infty}\frac{h(x)}{i(x)}$$

doesn't hold in general and to solve the LHS limit by l'Hopital, if necessary, we need to put it in the form

$$\lim_{x\to \infty}\left(\frac{f(x)}{g(x)} +\frac{h(x)}{i(x)}\right)=\lim_{x\to \infty}\frac{f(x)i(x)+h(x)g(x)}{g(x)i(x)} $$

the reason is that the case you are referring to is not among the cases considered by l'Hopital theorem.

user
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It seems to me you got some bad, or incomplete, advice. It's not incorrect if both of those limits exist and are finite. This doesn't have much to do with L'Hopital. The $\infty - \infty$ or $-\infty + \infty$ cases are problematic whether you're contemplating L'Hopital or not.

zhw.
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  • The advice was given HERE for that specific example and it was a completely different case to the one you are referring to. – user Nov 29 '18 at 07:41
  • @gimusi I responded to the OP's question here. She wrote "I was told that I cannot apply L'Hôpital's rule to each individual limit and then join the limits" and then asked why this is incorrect. I answered that. Why are you sending me to some other question? – zhw. Nov 29 '18 at 19:19
  • My aim was solely give you a better context for the question posed. – user Nov 29 '18 at 19:23