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I f space limit as x rightwards arrow 1 of fraction numerator x to the power of begin display style 1 over 100 end style end exponent minus 1 over denominator x to the power of k minus 1 end fraction equals 5 space t h e n space f i n d space t h e space v a l u e space o f space k ?

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Asked by Topperlearning User 16th November 2016, 1:54 AM
Answered by Expert
Answer:

limit as x rightwards arrow 1 of fraction numerator x to the power of begin display style 1 over 100 end style end exponent minus 1 over denominator x to the power of k minus 1 end fraction
equals limit as x rightwards arrow 1 of fraction numerator begin display style fraction numerator x to the power of 1 over 100 end exponent minus 1 over denominator x minus 1 end fraction end style over denominator begin display style fraction numerator x to the power of k minus 1 over denominator x minus 1 end fraction end style end fraction
U sin g space t h e space p r o p e r t y space limit as x rightwards arrow a of fraction numerator x to the power of n minus a to the power of n over denominator x minus a end fraction equals n a to the power of n minus 1 end exponent space w e space c a n space w r i t e
limit as x rightwards arrow 1 of fraction numerator begin display style fraction numerator x to the power of 1 over 100 end exponent minus 1 over denominator x minus 1 end fraction end style over denominator begin display style fraction numerator x to the power of k minus 1 over denominator x minus 1 end fraction end style end fraction equals fraction numerator begin display style 1 over 100 end style 1 to the power of begin display style fraction numerator negative 99 over denominator 100 end fraction end style end exponent over denominator k 1 to the power of k minus 1 end exponent end fraction equals fraction numerator 1 over denominator 100 k end fraction
G i v e n space t h a t space
limit as x rightwards arrow 1 of fraction numerator x to the power of begin display style 1 over 100 end style end exponent minus 1 over denominator x to the power of k minus 1 end fraction equals 5
rightwards double arrow fraction numerator 1 over denominator 100 k end fraction equals 5
rightwards double arrow k equals 1 over 500

Answered by Expert 16th November 2016, 3:54 AM
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