Rudin 8.5a

[SOLVED] rudin 8.5a

1. The problem statement, all variables and given/known data
Find the following limit:

[tex]\lim_{x \to 0} \frac{e-(1+x)^{1/x}}{x}[/tex]


2. Relevant equations
[tex]e = \lim_{x\to 0} (1+x)^{1/x}[/tex]

[tex]a^b = e^{b \log a}[/tex]


3. The attempt at a solution
Is this the derivative of something? I doubt it is but that is the only way I would know how to do this.

 

anyone?

 

Of course I know l'Hopital. Using it here requires me to calculate:

[tex]\lim_{x \to 0} \left(\frac{1}{x+x^2}-\frac{\log (1+x)}{x^2} \right) \exp \frac{\log (1+x)}{x}[/tex]

which I have no idea how to do. Of course I could probably use l'Hopital again but then it would just get messier. Of course I only need to find the limit of the expression in parenthesis but I am not really even sure if that exists.

 

Yay, I got it:

[tex]\lim_{x \to 0} \frac{(x/(1+x))-\log(x+1)}{x^2} = \lim_{x \to 0} \frac{1/(1+x)^2-1/(1+x)}{2x} = \lim_{x \to 0} \frac{-2/(1+x)^3+1/(1+x)^2}{2} = -1/2[/tex]

So the answer is e/2. Apparently lurflurf was right but I have no idea how he got those inequalities.


I think these computational problems are good preparation for the Putnam.