The function $f \colon x \mapsto (e^x-1)/x$ has singularity near $x = 0$. That singularity can be lifted, though: for $x = 1$, one should have $f(x) = 1$, since $$ e^x = \sum_{k=0} \frac{x^k}{k!} $$ and thus $$ (e^x - 1)/x = \sum_{k=1} \frac{x^{k-1}}{k!} $$ However, the form $(e^x-1)/x$ is not only not defined at $x = 0$, it is also numerically unstable in the vicinity of that point; in order to evaluate $f(x)$ for very small $x$ numerically, one could use a Taylor expansion, i.e. a truncation of the afore-mentioned power series.
Q: Does the function $f$ have a name? In other words, is this a common problem?
Q: Is anyone aware of a C/C++ library that handles this situation nicely, i.e. uses the Taylor expansion of an appropriate degree near 0 and the other representation away from zero?