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Find $\lim_{x\to \infty} \ln(e^{\operatorname{LambertW}(x)}+1)(e^{\operatorname{LambertW}(x)}+1) - x - \ln(x)$

Where the $LambertW$ function is defined here : http://en.wikipedia.org/wiki/Lambert_W

How to do this ?

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    I see no difference after the edit ? Only the + for infinity has been removed , which ironicly I would consider an edit if it was left out. Maybe I wrote find without capital. Im not complaining Im just curious what it was.2012-10-04
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    As for the suggestion of tag special function , that special function can be easily substituted away. So Im unsure about that.2012-10-04
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    The special function tag might attract other people to the question. Don't be mad, edits are made to help the OP as well as the readers. For example $$\lim_{x\rightarrow \infty} \big\{(e^{W(x)} + 1)\ln\big(e^{W(x)}+1\big) - x - \ln(x)\big\}$$ is far more readable.2012-10-04
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    @Pragabhava : Hmm the $e$ instead of $exp$ is worth consideration. But not everybody knows $W(x)$ although I could define it. But Im more concerned about leaving out the +.2012-10-04
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    @Pragabhava : Besides Im not mad. Extravert Psychology : Axiom 1 : 'not complaining' <-> not mad. Ok I made that up :)2012-10-04
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    One should add _where_ $W(x)$ _is the Lambert function_ (and maybe a link to Wikipedia). I didn't removed the +, just added the tag. If you ask me, is not missing.2012-10-04
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    What about $ln(x)$ for $x$ negative then ?2012-10-04
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    What about it?${}$2012-10-04
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    Its not real anymore ? And $LambertW(x)$ is not the same as $LambertW(-x)$ ?2012-10-04
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    $x\to\infty$ is short for $x\to+\infty$, and neither notation says anything about the domain including negatives.2012-10-04
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    Indeed, $x \rightarrow \infty$ means _there exist positive M such that_ $M< x$. Limits are _local_, your function is _local_, then it doesn't matter what happens away from the $\epsilon$-neighborhood.2012-10-04
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    Technically you guys are correct but I felt the + avoids confusion.2012-10-04

1 Answers 1

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Maple says $$ \lim_{x \to \infty}\left[\operatorname{e} ^{\bigl(LambertW (x) + 1\bigr)} \operatorname{ln} \biggl(\operatorname{e} ^{\bigl(LambertW (x) + 1\bigr)}\biggr) - x - \operatorname{ln} (x)\right] = \infty $$ if that's what you mean. In fact, $$ \lim_{x \to \infty}\frac{1}{x}\left[\operatorname{e} ^{\bigl(LambertW (x) + 1\bigr)} \operatorname{ln} \biggl(\operatorname{e} ^{\bigl(LambertW (x) + 1\bigr)}\biggr) - x - \operatorname{ln} (x)\right] = e-1 $$

edit Oct 4

OK, on this limit $$ \lim_{x\to\infty}\left[\Bigl(\operatorname{e} ^{W (x)} + 1\Bigr) \operatorname{ln} \Bigl(\operatorname{e} ^{W (x)} + 1\Bigr) - x - \operatorname{ln} (x)\right] $$ Maple says "Too many levels of recursion". So I used $e^{W(x)}=x/W(x)$ and then let $x=e^y$. (Also use: $\ln W(e^y) = y - W(e^y)$.) Now Maple says $$ \lim_{y\to\infty} \;\left[\frac{\operatorname{e} ^{y} \operatorname{ln} \bigl(\operatorname{e} ^{y} + W \bigl(\operatorname{e} ^{y}\bigr)\bigr)}{W \bigl(\operatorname{e} ^{y}\bigr)} - \frac{\operatorname{e} ^{y} y}{W \bigl(\operatorname{e} ^{y}\bigr)} + \operatorname{ln} \Bigl(\operatorname{e} ^{y} + W \bigl(\operatorname{e} ^{y}\bigr)\Bigr) - 2 y + W \bigl(\operatorname{e} ^{y}\bigr)\right] = -\infty $$ which is claimed as your answer...

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    No that is not what I meant. You got the +1 at the wrong position. Besides are you sure that is not a maple bug ?2012-10-04
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    Thanks. I might come back to this but this gets me going.2012-10-12