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?