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Question

Let a1>a2>a3>....>an>1.
p1>p2>p3>....>pn>0 such that p1+p2+p3+...+pn=1.
Also, F(x)=(p1ax1+p2ax2+...+pnaxn)1/x
limxF(x) equals

A
lnan
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B
ea1
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C
a1
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D
an
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Solution

The correct option is D an
Let limxF(x)=L
lnL=limxp1ax1lna1+p2ax2lna2+....+pnaxnlnanp1ax1+p2ax2+....+pnaxn
Dividing by (an)x and taking limx(a1an)x,(a2an)x, etc. vanish.
Therefore,
lnL=pnlnanpn
or L=an

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