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20 th sum limits

Asked by sawantsb765 | 04 Jun, 2019, 10:40: PM

Expert Answer

L H L equals limit as x rightwards arrow 0 to the power of minus of space f left parenthesis x right parenthesis equals limit as x rightwards arrow 0 of space fraction numerator 1 minus a to the power of x plus x space a to the power of x space ln space a over denominator a to the power of x space x squared end fraction equals limit as x rightwards arrow 0 of space fraction numerator 1 minus open parentheses 1 plus begin display style fraction numerator x ln space a over denominator 1 factorial end fraction end style plus begin display style fraction numerator open parentheses x space ln space a close parentheses squared over denominator 2 factorial end fraction end style close parentheses plus x space ln a open parentheses 1 plus begin display style fraction numerator x space ln space a over denominator 1 factorial end fraction end style close parentheses over denominator a to the power of x space x squared end fraction space.... space E x p a n d i n g space a to the power of x space a n d space n e g l e c t i n g space t h e space h i g h e r space p o w e r s equals limit as x rightwards arrow 0 of space fraction numerator 1 minus 1 minus begin display style fraction numerator x space ln space a over denominator 1 factorial end fraction end style minus begin display style fraction numerator open parentheses x space ln space a close parentheses squared over denominator 2 factorial end fraction end style plus x space ln a plus begin display style fraction numerator open parentheses x space ln space a close parentheses squared over denominator 1 factorial end fraction end style over denominator a to the power of x space x squared end fraction space equals limit as x rightwards arrow 0 of space fraction numerator begin display style open parentheses x space ln space a close parentheses squared over 2 end style over denominator a to the power of x space x squared end fraction space equals limit as x rightwards arrow 0 of space fraction numerator begin display style open parentheses ln space a close parentheses squared end style over denominator 2 space a to the power of x space end fraction space equals open parentheses ln space a close parentheses squared over 2 N o w comma space R H L equals limit as x rightwards arrow 0 to the power of plus of space f left parenthesis x right parenthesis equals limit as x rightwards arrow 0 of space fraction numerator 2 to the power of x space a to the power of x minus x space ln space 2 minus x space ln space a minus 1 over denominator x squared end fraction equals limit as x rightwards arrow 0 of space fraction numerator open parentheses 2 a close parentheses to the power of x minus x space ln space open parentheses 2 a close parentheses minus 1 over denominator x squared end fraction equals limit as x rightwards arrow 0 of space fraction numerator open parentheses 1 plus begin display style fraction numerator x space ln left parenthesis 2 a right parenthesis over denominator 1 factorial end fraction end style plus begin display style fraction numerator left parenthesis x space ln left parenthesis 2 a right parenthesis right parenthesis squared over denominator 2 factorial end fraction end style close parentheses minus x space ln space open parentheses 2 a close parentheses minus 1 over denominator x squared end fraction....... space E x p a n d i n g space left parenthesis 2 a right parenthesis to the power of x space a n d space n e g l e c t i n g space t h e space h i g h e r space p o w e r s equals limit as x rightwards arrow 0 of space fraction numerator begin display style fraction numerator left parenthesis x space ln left parenthesis 2 a right parenthesis right parenthesis squared over denominator 2 factorial end fraction end style over denominator x squared end fraction equals limit as x rightwards arrow 0 of space fraction numerator left parenthesis ln left parenthesis 2 a right parenthesis right parenthesis squared over denominator 2 end fraction equals fraction numerator left parenthesis ln left parenthesis 2 a right parenthesis right parenthesis squared over denominator 2 end fraction S i n c e comma space f left parenthesis x right parenthesis space i s space c o n t i n u o u s space a t space x equals 0 rightwards double arrow L H L equals R H L rightwards double arrow open parentheses ln space a close parentheses squared over 2 equals fraction numerator left parenthesis ln left parenthesis 2 a right parenthesis right parenthesis squared over denominator 2 end fraction rightwards double arrow left parenthesis ln space 2 a right parenthesis squared minus left parenthesis ln space a right parenthesis squared equals 0 rightwards double arrow left parenthesis ln space 2 a minus ln space a right parenthesis left parenthesis ln space 2 a plus ln space a right parenthesis equals 0 rightwards double arrow left parenthesis ln space 2 right parenthesis left parenthesis ln space 2 a squared right parenthesis equals 0 S i n c e space ln space 2 not equal to 0 rightwards double arrow ln space 2 a squared equals 0 rightwards double arrow 2 a squared equals 1 rightwards double arrow a equals fraction numerator 1 over denominator square root of 2 end fraction comma space a s space a greater than 0 f left parenthesis 0 right parenthesis equals open parentheses ln space begin display style fraction numerator 2 over denominator square root of 2 end fraction end style close parentheses squared over 2 equals open parentheses ln space square root of 2 close parentheses squared over 2 equals 1 half open parentheses 1 half ln square root of 2 close parentheses squared equals 1 over 8 open parentheses ln square root of 2 close parentheses squared

Answered by Renu Varma | 11 Jun, 2019, 12:35: PM

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