Are the following two limits the same?
$$\lim_{p_1 \to -\infty} \ln(1+(p_1)^2), \qquad (1)$$ and $$\lim_{p_2 \to \infty} \ln(1+(p_2)^2). \qquad (2)$$
I think they might be as the term which is being 'limited' is squared, and
$$\lim_{p \to -\infty} p^2 \equiv \lim_{p \to \infty} p^2,$$
right?
Yes, they are the same because $p^2=(-p)^2$ for all (real numbers) $p$. Also it should be easy to check that both limits tend to infinity.