Suppose that limx→∞cos(x)=L for some real number L. What this means is that, for every ?>0, there is a corresponding K∈R such that any x>K satisfies |cos(x)−L|<?. So let's take ?=12. No matter what K I pick, there's an x1>K such that cos(x1)=1 and an x2>K such that cos(x2)=−1. There's no L such that |1−L|<12 and |−1−L|<12 simultaneously, so the limit doesn't exist. Then cos(infinity) does not exist and cos(infinity) has no meaning.