Consider the function f(x)=ln(x)/√x
- Find the limit of f as x → 0 from the positive side.
- Find f(-1). Hint: f(-1)>0
1. As the natural log function decreases without limit (increases without limit in the negative direction) as its argument approaches zero from the positive side, and at the same time √x becomes closer to zero but still positive as x approaches zero, the function f(x)=ln(x)/√x also decreases without limit as x approaches zero, or in a less formal, looser terminology, approaches negative infinity.
2. In a famous equation, incorporating all the major building-block numbers and operations, e^(iπ)+1 = 0, or
e^(iπ)=-1 (by the way that π is a pi; it's sometimes hard to read in this font.)
As √(-1) is by definition i, we have iπ/i = π. The answer to the second part is pi.
Posted by Charlie
on 2003-07-01 03:00:44