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We put the 'fun' in function (Posted on 2003-07-01) Difficulty: 4 of 5
Consider the function f(x)=ln(x)/√x
  1. Find the limit of f as x → 0 from the positive side.
  2. Find f(-1). Hint: f(-1)>0

See The Solution Submitted by Bryan    
Rating: 4.0000 (2 votes)

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Solution solution | Comment 1 of 4
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.)
or
ln(-1)=iπ

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
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