Letting r be the radius of circle C2 and using a little algebra and
geometry we get
r2
|BG| = -------------------
2 - sqrt(4 - r2)
Therefore,
r2
limit |BG| = limit -------------------
r->0 r->0 2 - sqrt(4 - r2)
Plugging r=0 into this gives 0/0. So using the rule of L'Hospital gives
d(r2)/dr
limit |BG| = limit -------------------------
r->0 r->0 d(2 - sqrt(4 - r2))/dr
= limit 2*sqrt(4 - r2)
r->0
= 4
A simpler approach is to multiply numerator and
denominator of
r2
|BG| = -------------------
2 - sqrt(4 - r2)
by 2 + sqrt(4 - r2) to get
|BG| = 2 + sqrt(4 - r2)
Thus,
limit |BG| = 4
r->0
A different approach:
Let t = angle BCE = angle BEC and p = angle ABE = angle AEB. Then
|BG| = r*tan(t)
r = 2*cos(p)
p = 2*t - 90
Combining these we get
|BG| = 4*sin2(t)
As r->0, t->90. Therefore,
limit |BG| = 4
t->90
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