lim _(xarrow 0)(ln(1+x))/(3x)

To evaluate the limit \(\lim_{x \rightarrow 0} \frac{\ln(1+x)}{3x}\), we can use the fact that \(\ln(1+x)\) can be expanded into a Taylor series around . The Taylor series expansion for \(\ln(1+x)\) is: We only need the first-order term because higher-order terms will become negligible as approaches 0. Therefore, we can approximate \(\ln(1+x)\) by near : Substituting this approximation into the limit, we get: The terms cancel out: