A contradiction I found with integrals and limits of an exponential function "a" is a constant.
$$\lim_{a\to 0^+} \int e^{ax} \, dx = \lim_{a\to 0^+} \frac{e^{ax}}{a}=\frac{e^{0^+\times x}}{0^+}=\frac{1}{0^+}=\infty$$
but doing it in a different order:
$$\lim_{a\to 0^+} \int e^{ax} \, dx =\int e^{0^+\times x} \, dx=\int e^{0} \, dx=\int 1 \, dx=x$$
so which one is it? why did I get this contradiction? Thanks.

It should be
$$
\int e^{ax}dx = \frac{e^{ax}}{a}+C.
$$
$C$ is an arbitrary constant and can always be chosen as $-\frac{1}{a}+C'$. So, you can see that both limits coincide.
