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If lim(xtoa)(a^(x)-x^(a))/(x^(x)-a^(a))=...

If `lim_(xtoa)(a^(x)-x^(a))/(x^(x)-a^(a))=-1" and "agt0," then find the value of a."`

A

43529

B

2

C

`"sin"(2)/(5)`

D

`"sin"(1)/(5)`

Text Solution

Verified by Experts

The correct Answer is:
`a=1`
Since `underset(xtoa)lim(a^(x)-x^(a))/(x^(x)-a^(a))=-1" ,`
`underset(xtoa)lim(a^(x)loga-ax^(a-1))/(x^(x)(1+logx))=-1`
or `underset(xtoa)lim(a^(a)[loga-1])/(a^(a)[1+loga])=-1`
or` loga-1=-1-loga`
or` 2loga=0`
or` loga=0`
or` a=1`
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