Improper Integrals - infinity The improper integral is defined as: $$\int_{-\infty}^{\infty} f(x)\,\mathrm{d}x= \int_{-\infty}^{a}f(x)\,\mathrm{d}x+\int_{a}^{\infty}f(x)\,\mathrm{d}x$$
And we also have $\lim_{N\to\infty} \int_{-N}^{a}f(x)\,\mathrm{d}x = \int_{-\infty}^{a}f(x)\,\mathrm{d}x$
The integral $\int_{-\infty}^{\infty} x\,\mathrm{d}x$ does not exist because $\int_{-\infty}^{a}x\,\mathrm{d}x$ does not exist. 
But the integral $\lim_{N\to\infty} \int_{-N}^{N} x\,\mathrm{d}x$ does exist since $\lim_{N \to \infty} [N^2/2 - N^2/2] = 0$
But can we not also define it as $$\lim_{N\to\infty} \int_{-N}^{N} x\,\mathrm{d}x = \lim_{N\to\infty} \int_{-N}^{a} x\,\mathrm{d}x+ \lim_{N\to\infty} \int_{a}^{N} x\,\mathrm{d}x=\int_{-\infty}^{a}x\,\mathrm{d}x + \int_{a}^{\infty}x\,\mathrm{d}x$$
Now doesn't this make the LHS does not exist?

Your third statement is incorect,
 $\lim_{N\to\infty} \int_{-N}^{N} x$ does not exist.
And the reason is your last statement.
$\int_{-\infty}^{\infty} x = lim_{a\to\infty}lim_{b\to\infty} \int_{-a}^{b} x$
your last statement is also a way to define $\int_{-\infty}^{\infty} x$ and they both coreect 
and the integral does not Exist.
If you have to integrate  odd function between finite  symmetrical section your integral will be zero but in infinity section you cant know how your function will behave.
