Could someone explain why our signals are not always band limited?
The triangle in the hint stands out because of its sharp and clearly defined edges. I.e. the edges of the triangle are formed from a discontinuous change in the image intensity from white to red. This discontinuous and abrupt change in intensity corresponds to high frequencies (since a high frequency is after all, a high rate of change). In particular, signals with such discontinuities have an infinite frequency spectrum. Hence, they are not band-limited (or constrained to a band of low frequencies).