No idea if this is the right stackexchange for this, feel free to point me elsewhere!
I'm teaching business calculus and one of the problems the students have is to figure out the present value of a continuous income stream $\$$$A$ per year earning $r$ interest rate. The integral comes to $$P.V=A\int_0^te^{-rz}dz=\frac A {-r}(e^{-rt}-1)$$ I've noticed as a heuristic that for relatively short values of t ($t<10$) and small values of r ($r<.04$) (the parameters the learning software uses), the present value comes to be just a bit less than $A\cdot t$. (usually within about 10 percent) Does anyone know an economics rational why this is the case, and would it hold for larger values of $r$ or $t$?