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Asked by sarveshvibrantacademy | 05 May, 2019, 15:51: PM
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limit as n rightwards arrow infinity of space n squared open parentheses a to the power of 1 divided by n end exponent minus a to the power of 1 divided by left parenthesis n plus 1 right parenthesis end exponent close parentheses equals limit as n rightwards arrow infinity of space a to the power of 1 divided by n end exponent space space n squared open parentheses 1 minus a to the power of 1 divided by left parenthesis n plus 1 right parenthesis minus 1 divided by n end exponent close parentheses space.... space T a k i n g space a to the power of 1 divided by n end exponent space c o m m o n equals limit as n rightwards arrow infinity of space a to the power of 1 divided by n end exponent space space n squared open parentheses 1 minus a to the power of negative 1 divided by n left parenthesis n plus 1 right parenthesis end exponent close parentheses equals limit as n rightwards arrow infinity of space a to the power of 1 divided by n end exponent space space n squared open parentheses fraction numerator a to the power of 1 divided by n left parenthesis n plus 1 right parenthesis end exponent minus 1 over denominator a to the power of 1 divided by n left parenthesis n plus 1 right parenthesis end exponent end fraction close parentheses equals limit as n rightwards arrow infinity of space a to the power of 1 divided by n end exponent over a to the power of 1 divided by n left parenthesis n plus 1 right parenthesis end exponent space space n space open parentheses fraction numerator a to the power of 1 divided by n left parenthesis n plus 1 right parenthesis end exponent minus 1 over denominator begin display style 1 over n end style end fraction close parentheses equals limit as n rightwards arrow infinity of space a to the power of 1 divided by n end exponent over a to the power of 1 divided by n left parenthesis n plus 1 right parenthesis end exponent space space open parentheses fraction numerator n over denominator n plus 1 end fraction close parentheses space open parentheses fraction numerator a to the power of 1 divided by n left parenthesis n plus 1 right parenthesis end exponent minus 1 over denominator begin display style fraction numerator 1 over denominator n left parenthesis n plus 1 right parenthesis end fraction end style end fraction close parentheses equals limit as n rightwards arrow infinity of space a to the power of begin display style 1 over n end style end exponent over a to the power of fraction numerator 1 over denominator n left parenthesis n plus 1 right parenthesis end fraction end exponent space space open parentheses fraction numerator n over denominator n plus 1 end fraction close parentheses space open parentheses fraction numerator a to the power of fraction numerator 1 over denominator n left parenthesis n plus 1 right parenthesis end fraction end exponent minus 1 over denominator begin display style fraction numerator 1 over denominator n left parenthesis n plus 1 right parenthesis end fraction end style end fraction close parentheses equals limit as n rightwards arrow infinity of space a to the power of fraction numerator 1 over denominator n plus 1 end fraction end exponent space space open parentheses fraction numerator n over denominator n plus 1 end fraction close parentheses space open parentheses fraction numerator a to the power of fraction numerator 1 over denominator n left parenthesis n plus 1 right parenthesis end fraction end exponent minus 1 over denominator begin display style fraction numerator 1 over denominator n left parenthesis n plus 1 right parenthesis end fraction end style end fraction close parentheses equals limit as n rightwards arrow infinity of space a to the power of fraction numerator 1 over denominator n plus 1 end fraction end exponent space space open parentheses fraction numerator 1 over denominator 1 plus begin display style 1 over n end style end fraction close parentheses space open parentheses fraction numerator a to the power of fraction numerator 1 over denominator n left parenthesis n plus 1 right parenthesis end fraction end exponent minus 1 over denominator begin display style fraction numerator 1 over denominator n left parenthesis n plus 1 right parenthesis end fraction end style end fraction close parentheses equals a to the power of 1 over infinity end exponent open parentheses fraction numerator 1 over denominator 1 plus begin display style 1 over infinity end style end fraction close parentheses space ln space a space.......... space open square brackets S i n c e space limit as x rightwards arrow 0 of space open parentheses fraction numerator a to the power of x minus 1 over denominator x end fraction close parentheses equals ln space a space a n d space sin c e space n rightwards arrow infinity space t h e r e f o r e space fraction numerator 1 over denominator n left parenthesis n plus 1 right parenthesis end fraction rightwards arrow 0 close square brackets equals a to the power of 0 space ln space a space equals ln space a H e n c e comma space limit as n rightwards arrow infinity of space n squared open parentheses a to the power of 1 divided by n end exponent minus a to the power of 1 divided by left parenthesis n plus 1 right parenthesis end exponent close parentheses equals ln space a
Answered by Renu Varma | 07 May, 2019, 10:13: AM

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