"infinity" and "negative infinity" are NOT numbers in the usual real number system (there are number systems that do include them, but that is much more advanced). When we say "the limit is infinity" or "the limit is negative infinity" we mean "the limit doe not exist"- in a specific way. Your statement "if something goes toward positive or negative infinity, then the area of the curve cannot be measured" is too vague. If what goes toward infinity? The graph of \(\displaystyle y= 1/x^2\) has x going to infinity but the area under \(\displaystyle y= 1/x^2\), above y= 0, x from 1 to infinity, is finite: 1. On the other hand, \(\displaystyle y= x^{-1/2}\) has y going to "infinity" as x goes to 0 but the region bounded by \(\displaystyle y= x^{-1/2}\), y= 0, x= 0, and x= 1 is finite, 2.Do such limits exists? It seems from my book that they don't. In other words, if something goes toward positive or negative infinity, then the area of the curve cannot be measured. On the other hand, curves with limits (found by solving for definite integrals) have areas that can be measured (though not absolutely).