- #1

Jezza

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## Homework Statement

[tex]\lim\limits_{x \to 0} \left(\ln(1+x)\right)^x[/tex]

## Homework Equations

Maclaurin series:

[tex]\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + ... + (-1)^{r+1} \frac{x^r}{r} + ...[/tex]

## The Attempt at a Solution

We're considering vanishingly small [itex]x[/itex], so just taking the first term in the Maclaurin series the limit becomes:

[tex]\lim\limits_{x \to 0} \left(\ln(1+x)\right)^x = \lim\limits_{x \to 0} x^x = \mathrm{undefined}[/tex]

or so I thought until google tells me that [itex]0^0 = 1[/itex].

What's going on here? How can I evaluate the limit properly?