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Let P(n)=a^(P(n-1))-1,AA n=2,3,..., and ...

Let `P_(n)=a^(P_(n-1))-1,AA n=2,3,...,` and let `P_(1)=a^(x)-1,` where `ainR^(+).` Then evaluate `lim_(xto0) (P_(n))/(x).`

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Clearly, if `P_(k)to0,` then `P_(k+1)to0.`
Now, as `xto0,` we get `P_(1)to0` or `P_(2),P_(3),P_(4),…,P_(n)to0`. Therefore,
`underset(xto0)lim(P_(n))/(x)=underset(xto0)lim(P_(n))/(P_(n-1))(P_(n-1))/(P_(n-2))...(P_(1))/(x)`
Now, `underset(xto0)lim(P_(k))/(P_(k-1))=underset(xto0)lim(a^(P_(k-1))-1)/(P_(k-1)) =1n" "a`
`:. " "" Required limit "=(ln" "a)^(n)`
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