Why are the improper integrals
$$\int_{-\infty}^{\infty}x\,\mathrm{d}x \qquad\text{and}\qquad \lim_{n\to\infty} \int_{-n}^{n}x\,\mathrm{d}x$$
different?
This is question #11.4 from the 2015 ANPEC Exam. It's originally presented with an equality and the official answer is "False". There is no further context on the solution, so I'd like some help with the intuition behind this conclusion.
The full exam can be accessed here.
Best Answer
Because the left integral, by definition, diverges, whereas the right limit is equal to zero.
The definition for the left integral, as is probably cited in the your textbook, is given by
$$\int_{-\infty}^\infty x \, \mathrm{d} x := \lim_{a \to -\infty} \int_a^c x \, \mathrm{d} x + \lim_{b \to \infty} \int_c^b x \, \mathrm{d} x$$
for any $c$ of your choice satisfying $-\infty < c < \infty$. For this integral to exist, both of these limits must exist; however, neither is finite for this limit, and it is easy to see why.
Loosely, I like to think of this as meaning that you need to account for all possible ways or speeds at which the bounds can go off to infinity. For instance, it is fine to say
$$\int_{-\infty}^\infty f(x) \, \mathrm{d} x = \lim_{\substack{a \to -\infty \\ b \to +\infty}} \int_a^b f(x) \, \mathrm{d} x$$
because this handles that. Letting $a=-b$ however is restricting on this and forces them to go to infinity at the same rate.