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JEE Class main Answered

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Asked by sarveshvibrantacademy | 02 May, 2019, 12:09: AM
Expert Answer
begin mathsize 16px style limit as straight x rightwards arrow infinity of space open parentheses fraction numerator straight a subscript 1 superscript begin display style 1 over straight x end style end superscript plus straight a subscript 2 superscript begin display style 1 over straight x end style end superscript plus straight a subscript 3 superscript begin display style 1 over straight x end style end superscript plus..... plus straight a subscript straight n superscript begin display style 1 over straight x end style end superscript over denominator straight n end fraction close parentheses to the power of nx
Put space 1 over straight x equals straight t rightwards double arrow straight x equals 1 over straight t
When space straight x rightwards arrow infinity space then space straight t rightwards arrow 0
equals limit as straight t rightwards arrow 0 of space open parentheses fraction numerator straight a subscript 1 superscript begin display style straight t end style end superscript plus straight a subscript 2 superscript begin display style straight t end style end superscript plus straight a subscript 3 superscript begin display style straight t end style end superscript plus..... plus straight a subscript straight n superscript begin display style straight t end style end superscript over denominator straight n end fraction close parentheses to the power of straight n over straight t end exponent
equals limit as straight t rightwards arrow 0 of space open parentheses fraction numerator straight a subscript 1 superscript begin display style straight t end style end superscript minus 1 plus straight a subscript 2 superscript begin display style straight t end style end superscript minus 1 plus straight a subscript 3 superscript begin display style straight t end style end superscript minus 1 plus..... plus straight a subscript straight n superscript begin display style straight t end style end superscript minus 1 plus straight n over denominator straight n end fraction close parentheses to the power of straight n over straight t end exponent
equals limit as straight t rightwards arrow 0 of space open parentheses fraction numerator straight a subscript 1 superscript begin display style straight t end style end superscript minus 1 over denominator straight n end fraction plus fraction numerator straight a subscript 2 superscript begin display style straight t end style end superscript minus 1 over denominator straight n end fraction plus fraction numerator straight a subscript 3 superscript begin display style straight t end style end superscript minus 1 over denominator straight n end fraction plus..... plus fraction numerator straight a subscript straight n superscript begin display style straight t end style end superscript minus 1 over denominator straight n end fraction plus 1 close parentheses to the power of straight n over straight t end exponent
equals limit as straight t rightwards arrow 0 of space open parentheses straight t over straight n open square brackets fraction numerator straight a subscript 1 superscript begin display style straight t end style end superscript minus 1 over denominator straight t end fraction plus fraction numerator straight a subscript 2 superscript begin display style straight t end style end superscript minus 1 over denominator straight t end fraction plus fraction numerator straight a subscript 3 superscript begin display style straight t end style end superscript minus 1 over denominator straight t end fraction plus..... plus fraction numerator straight a subscript straight n superscript begin display style straight t end style end superscript minus 1 over denominator straight t end fraction close square brackets plus 1 close parentheses to the power of straight n over straight t end exponent
equals limit as straight t rightwards arrow 0 of space open parentheses straight t over straight n open parentheses log space straight a subscript 1 plus log space straight a subscript 2 plus log space straight a subscript 3 plus..... plus log space straight a subscript straight n close parentheses plus 1 close parentheses to the power of straight n over straight t end exponent
equals 1 end style
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