# Problem with limit

• September 5th 2009, 07:54 PM
atlantis84
Problem with limit
Any help to the following?

$\lim_{x \rightarrow 0} \left(\frac{ln (x+1) }{x} \right)^{\frac{2}{X}}$
• September 5th 2009, 10:42 PM
mr fantastic
Quote:

Originally Posted by atlantis84
Any help to the following?

$\lim_{x \rightarrow 0} \left(\frac{ln (x+1) }{x} \right)^{\frac{2}{X}}$

One possible approach:
1. Substitute the following Maclaurin series: $\ln (x + 1) = x - \frac{x^2}{2} + \frac{x^3}{3} - ....$.

2. Simplify the result.

3. Consider $\lim_{x \rightarrow 0} e^{\ln \left( 1 - \frac{x}{2} + \frac{x^2}{3} - .... \right)^{2/x}} = \lim_{x \rightarrow 0} e^{2 \ln \left( 1 - \frac{x}{2} + \frac{x^2}{3} - .... \right)/x}$.

4. Consider the limit of the exponent using l'Hopital's rule.

Alternatively, there may be a clever substitution that can first be made.