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Re: Problems with Limit, Log, E

Robert Lawrence wrote:
> When I attempt to evaluate the expresssion
>          Limit[ x - Log[1 + E^x],  x -> Infinity]
> I simply get back the original expression.  It seems to me that this
> should evaluate to zero.
> For that matter, when I evaluate the expression
>         Log[E^x]
> I also get back the original expression, instead of x.
> I would appreciate any thoughts concerning these results.
> I am running Mathematica 3.0.1 under Windows 95.
> Robert Lawrence

1)  Mathematica is a programming language and is far different in
concept from "mathematical scratchpad" programs like MathCad.  Entering
an expression like 


is not supposed to return anything.  You haven't told the computer you
wanted it to do anything with your expression.  In your case, I guess
that you wanted it to somehow simplify your expression.  In the general
case, Log[Exp[x]] is multi-valued for complex x (Log is only the
inverse of Exp for real arguments), so 

ComplexExpand[Log[E^x]]  is the command for you.

2)  Because, x-Log[1+Exp[x]] is a difference, there is no way to do it
symbolically as it stands.  L'Hopitals rule doesn't apply, so
Mathematica returns it to you.  If you were going to evaluate it by
hand, you would have to manipulate it a bit,  as in:


The limit of Log[1+Exp[x]]/x is 1 as x approaches Infinity, so the
entire expression is still indeterminate.

The next step is to rewrite it:

(1-Log[1-Exp[x]]/x) / (1/x)

so you can apply L'Hopitals rule.   A couple of tries at this and you
will find that the derivatives get really ugly and keep on getting more
and more complex so L'Hopitals rule, while true, is not very helpful. 
Mathematica must not have any secret methods since putting in the above
formulas merely makes the algorithm think for a couple of minutes
before spitting the original expression back out.  Congratulations! 
You found a really nasty limit!

Fortunately, the simple-minded Engineers method is simply to graph the
function from 1 to 1000 and see what the trend is: Plot[x
and you will see that the limit is zero.  This probably wouldn't satisfy
a real mathematician, so, If your goal is to prove the limit, get ready
for a lot of messy and creative algebra, but if you only wanted to know
what the limit was, its pretty easy.

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