A thought I had to resolve it simply, is to just accept that 1/x doesn't actually reach 0, so therefore ln(x) will be increasing bysomething, even if that something is small. But then I thought, even though 1/x doesn't actually reach 0, the rate of change in some sense converges to 0 for ln(x), so ln(x) must converge then to some value, no?

Just as we're taught the infinite sequence 1 + 1/n for n to infinity converges to 2?

But we know from a number of proofs already, that ln(x) diverges to infinity, so I'm still stuck on explaining that in relation to what I posed above.