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Asked by sarveshvibrantacademy | 12 May, 2019, 14:23: PM
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f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell x left square bracket x right square bracket squared log subscript left parenthesis 1 plus x right parenthesis end subscript space 2 comma space f o r space minus 1 less than x less than 0 end cell row cell fraction numerator ln open parentheses e to the power of x squared end exponent plus 2 square root of left curly bracket x right curly bracket end root close parentheses over denominator tan square root of x end fraction comma space f o r space 0 less than x less than 1 end cell end table close S i n c e comma space left square bracket x right square bracket equals negative 1 space f o r space minus 1 less than x less than 0 space a n d space left curly bracket x right curly bracket equals x minus left square bracket x right square bracket equals x minus 0 equals x space b e c a u s e space left square bracket x right square bracket equals 0 space f o r space 0 less than x less than 1 rightwards double arrow f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell x space log subscript left parenthesis 1 plus x right parenthesis end subscript space 2 comma space f o r space minus 1 less than x less than 0 end cell row cell fraction numerator ln open parentheses e to the power of x squared end exponent plus 2 square root of x close parentheses over denominator tan square root of x end fraction comma space f o r space 0 less than x less than 1 end cell end table close N o w comma space c o n s i d e r space t h e space L H L 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 x space log subscript left parenthesis 1 plus x right parenthesis end subscript space 2 equals limit as x rightwards arrow 0 of space log subscript left parenthesis 1 plus x right parenthesis end subscript space 2 to the power of x equals limit as x rightwards arrow 0 of space fraction numerator log space 2 to the power of x over denominator log left parenthesis 1 plus x right parenthesis end fraction space space rightwards arrow space 0 over 0 space f o r m space a s space x space t e n d s space t o space 0 equals limit as x rightwards arrow 0 of space fraction numerator begin display style 1 over 2 to the power of x end style times begin display style fraction numerator d over denominator d x end fraction end style open parentheses 2 to the power of x close parentheses over denominator begin display style fraction numerator 1 over denominator left parenthesis 1 plus x right parenthesis end fraction end style end fraction space.... space U sin g space L apostrophe H o s p i t a l apostrophe s space r u l e equals limit as x rightwards arrow 0 of space left parenthesis 1 plus x right parenthesis times 1 over 2 to the power of x times open parentheses 2 to the power of x space ln space 2 close parentheses equals limit as x rightwards arrow 0 of space left parenthesis 1 plus x right parenthesis times ln space 2 equals ln space 2 N o w comma space c o n s i d e r space t h e space R H L 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 fraction numerator ln open parentheses e to the power of x squared end exponent plus 2 square root of x close parentheses over denominator tan square root of x end fraction space space rightwards arrow space 0 over 0 space f o r m space a s space x space t e n d space t o space 0 equals limit as x rightwards arrow 0 of space fraction numerator begin display style fraction numerator 1 over denominator e to the power of x squared end exponent plus 2 square root of x end fraction end style times begin display style fraction numerator d over denominator d x end fraction end style open parentheses e to the power of x squared end exponent plus 2 square root of x close parentheses over denominator s e c squared square root of x times begin display style fraction numerator d over denominator d x end fraction end style open parentheses square root of x close parentheses end fraction equals limit as x rightwards arrow 0 of space fraction numerator begin display style fraction numerator 1 over denominator e to the power of x squared end exponent plus 2 square root of x end fraction times open parentheses e to the power of x squared end exponent space times 2 x plus fraction numerator 1 over denominator square root of x end fraction close parentheses end style over denominator s e c squared square root of x times begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style end fraction equals limit as x rightwards arrow 0 of space fraction numerator begin display style fraction numerator 1 over denominator e to the power of x squared end exponent plus 2 square root of x end fraction times open parentheses fraction numerator e to the power of x squared end exponent space times 2 x square root of x plus 1 over denominator square root of x end fraction close parentheses end style over denominator s e c squared square root of x times begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style end fraction equals limit as x rightwards arrow 0 of space fraction numerator 2 open parentheses e to the power of x squared end exponent space times 2 x square root of x plus 1 close parentheses over denominator open parentheses e to the power of x squared end exponent plus 2 square root of x close parentheses s e c squared square root of x end fraction equals 2 not equal to R H L H e n c e comma space f space i s space d i s c o n t i n u o u s space a t space x equals 0 space a n d space h e n c e space h a s space i r r e m o v a b l e space d i s c o n t i n u i t y space a t space x equals 0.
Answered by Renu Varma | 15 May, 2019, 11:20: AM

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