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Asked by sarveshvibrantacademy | 09 May, 2019, 23:42: PM
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f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell sin open parentheses 5 over x close parentheses comma space x not equal to 0 end cell row cell 0 comma space space space space space space space space space space x equals 0 end cell end table close C l e a r l y comma space f left parenthesis x right parenthesis space i s space c o n t i n u o u s f left parenthesis g left parenthesis x right parenthesis right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell sin open parentheses fraction numerator 5 over denominator g left parenthesis x right parenthesis end fraction close parentheses comma space g left parenthesis x right parenthesis not equal to 0 end cell row cell 0 comma space space space space space space space space space space space space space space space g left parenthesis x right parenthesis equals 0 end cell end table close G i v e n space t h a t space f left parenthesis g left parenthesis x right parenthesis right parenthesis space i s space c o n t i n u o u s rightwards double arrow limit as g left parenthesis x right parenthesis rightwards arrow 0 of f left parenthesis g left parenthesis x right parenthesis right parenthesis equals f left parenthesis 0 right parenthesis rightwards double arrow sin open parentheses fraction numerator 5 over denominator g left parenthesis x right parenthesis end fraction close parentheses equals 0 rightwards double arrow fraction numerator 5 over denominator g left parenthesis x right parenthesis end fraction equals 0 rightwards double arrow g left parenthesis x right parenthesis equals 0 space f o r space x rightwards arrow 0 space sin c e space 5 not equal to 0 N o w comma space l e t apostrophe s space c h e c k space t h e space g i v e n space o p t i o n s colon A right parenthesis I f space g left parenthesis x right parenthesis equals x squared plus 1 comma space x element of R rightwards double arrow g left parenthesis x right parenthesis greater or equal than 1 space f o r space a l l space x element of R rightwards double arrow g left parenthesis x right parenthesis not equal to 0 space f o r space x equals 0 S o comma space g left parenthesis x right parenthesis equals x squared plus 1 space i s space n o t space p o s s i b l e B right parenthesis I f space g left parenthesis x right parenthesis equals ln left parenthesis 1 plus x right parenthesis comma space x element of left parenthesis negative 1 comma infinity right parenthesis rightwards double arrow g left parenthesis x right parenthesis equals ln space 1 equals 0 space f o r space x equals 0 space a n d space i t space i s space d e f i n e d space f o r space o t h e r space v a l u e s space o f space x element of left parenthesis negative 1 comma infinity right parenthesis S o comma space g left parenthesis x right parenthesis equals ln left parenthesis 1 plus x right parenthesis space i s space p o s s i b l e C right parenthesis I f space g left parenthesis x right parenthesis equals cos space x comma space x element of R rightwards double arrow g left parenthesis x right parenthesis equals 0 space f o r space space x equals straight pi over 2 rightwards double arrow space f left parenthesis g left parenthesis x right parenthesis right parenthesis space i s space u n d e f i n e d space f o r space x equals straight pi over 2 space a n d space g left parenthesis 0 right parenthesis equals 1 not equal to 0 S o comma space g left parenthesis x right parenthesis equals cos space x space i s space n o t space p o s s i b l e D right parenthesis I f space g left parenthesis x right parenthesis equals e to the power of x comma space x element of R rightwards double arrow g left parenthesis x right parenthesis equals 0 space f o r space space x equals negative infinity rightwards double arrow space f left parenthesis g left parenthesis x right parenthesis right parenthesis space i s space u n d e f i n e d space f o r space x equals negative infinity space a n d space g left parenthesis 0 right parenthesis equals 1 not equal to 0 S o comma space g left parenthesis x right parenthesis equals e to the power of x space i s space n o t space p o s s i b l e H e n c e comma space left parenthesis f g right parenthesis 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 space i f space g left parenthesis x right parenthesis equals ln left parenthesis 1 plus x right parenthesis.
Answered by Renu Varma | 10 May, 2019, 12:18: PM

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