# How do you evaluate the limit (1/(1+r)-1)/r as r approaches 0?

Oct 2, 2016

$\frac{\frac{1}{1 + r} - 1}{r}$

$= \frac{\frac{1}{1 + r} - \frac{1 + r}{1 + r}}{r}$

$= \frac{\frac{1 - \left(1 + r\right)}{1 + r}}{r}$

$= \frac{\frac{1 - 1 - r}{1 + r}}{r}$

$= \frac{- \frac{r}{1 + r}}{r}$

$= - \frac{r}{1 + r} \cdot \frac{1}{r}$

$= - \frac{1}{1 + r}$

Now:

${\lim}_{r \to 0} - \frac{1}{1 + r} = - 1$