I have the following integral
$$\int_{- \infty}^\infty e^{-|x|} dx$$
and the following two questions
(1) Since the preimages $x$ determine the the images $e^{-|x|}$ for nonnegative and negative preimages, I was wondering if splitting the integral into two integrals with the following limits of integration is correct
$$\lim \limits_{c \rightarrow 0^-} \enspace \int_{- \infty}^{c} e^{-|x|} \enspace dx \quad + \quad \int_{0}^{\infty} e^{- |x|} \enspace dx$$
where $c$ obviously approaches $0$ from below without necessarily equalling $0$, so that $e^{- |x|} = e{-(-x)}$. So,
$$\lim \limits_{c \rightarrow 0^-} \enspace \int_{- \infty}^{c} e^{-|x|} \enspace dx \quad + \quad \int_{0}^{\infty} e^{- |x|} \enspace dx $$
$$= \lim \limits_{c \rightarrow 0^-} \enspace \int_{- \infty}^{c} e^{x} \enspace dx \quad + \quad \int_{0}^{\infty} e^{-x} \enspace dx $$
$$= 2$$
So I was wondering if the above integrals is correct, especially with the use of the limit; and then,
(2) Is the following integral incorrect, because I think it is
$$\int_{- \infty}^{0} e^{-|x|} \enspace dx \quad + \quad \int_{0}^{\infty} e^{- |x|} \enspace dx$$
where my issue is with the upper limit of integration $0$ of the first integral, since the $x$ spans negative values and $0$, causing problems with the integrand due to the absolute value.
Also,I know that the function $f$ is even, so it is better to use the following identity in order for my problem to be easily avoided
$$\int_{-\infty}^{\infty} f(x) \enspace dx = 2 \int_0^{\infty} f(x) \enspace$$
but I'm very curious to see an answer to my questions above.