Question
Sun April 19, 2015 By: Shivani P

limit x-->0 [((log(1+x)^(1+x))/x^2 )-(1/x) 
options:
a] 1/2
b] -1/2
c] 0
d] 1

Expert Reply
Vimala Ramamurthy
Mon April 20, 2015
C o n s i d e r space t h e space g i v e n space e x p r e s s i o n colon limit as x rightwards arrow 0 of fraction numerator log open parentheses 1 plus x close parentheses to the power of 1 plus x end exponent over denominator x squared end fraction minus 1 over x equals limit as x rightwards arrow 0 of fraction numerator open parentheses 1 plus x close parentheses log open parentheses 1 plus x close parentheses over denominator x squared end fraction minus 1 over x equals limit as x rightwards arrow 0 of fraction numerator open parentheses 1 plus x close parentheses open square brackets x minus begin display style x squared over 2 end style plus begin display style x cubed over 3 end style minus begin display style x to the power of 4 over 4 end style plus begin display style x to the power of 5 over 5 end style minus... close square brackets over denominator x squared end fraction minus 1 over x equals limit as x rightwards arrow 0 of fraction numerator open parentheses 1 plus x close parentheses open square brackets x minus begin display style x squared over 2 end style plus begin display style x cubed over 3 end style minus begin display style x to the power of 4 over 4 end style plus begin display style x to the power of 5 over 5 end style minus... close square brackets minus x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator open square brackets x minus begin display style x squared over 2 end style plus begin display style x cubed over 3 end style minus begin display style x to the power of 4 over 4 end style plus begin display style x to the power of 5 over 5 end style minus... close square brackets plus x open square brackets x minus x squared over 2 plus x cubed over 3 minus x to the power of 4 over 4 plus x to the power of 5 over 5 minus... close square brackets minus x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator open square brackets negative begin display style x squared over 2 end style plus begin display style x cubed over 3 end style minus begin display style x to the power of 4 over 4 end style plus begin display style x to the power of 5 over 5 end style minus... close square brackets plus open square brackets x squared minus x cubed over 2 plus x to the power of 4 over 3 minus x to the power of 5 over 4 plus x to the power of 6 over 5 minus... close square brackets over denominator x squared end fraction equals limit as x rightwards arrow 0 of open curly brackets open square brackets negative 1 half plus x over 3 minus x squared over 4 plus x cubed over 5 minus... close square brackets plus open square brackets 1 minus x over 2 plus x squared over 3 minus x cubed over 4 plus x to the power of 4 over 5 minus... close square brackets close curly brackets equals 1 minus 1 half equals 1 half H e n c e space left parenthesis a right parenthesis space i s space t h e space c o r r e c t space o p t i o n.
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