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JEE Class main Answered

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Asked by sarveshvibrantacademy | 17 May, 2019, 09:33: AM
answered-by-expert Expert Answer
f left parenthesis x right parenthesis space c a n space b e space r e w r i t t e n space a s
f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell b subscript n plus cos pi x comma space space space f o r space x element of left parenthesis 2 n minus 1 comma space 2 n right parenthesis end cell row cell a subscript n plus sin pi x comma space space space space f o r space x element of left square bracket 2 n comma space 2 n plus 1 right square bracket space end cell end table close
A t space x equals 2 n
F o r space l e f t space h a n d space l i m i t comma space l e t space x equals 2 n minus h comma space a s space x rightwards arrow 2 n comma space h rightwards arrow 0
L H L equals limit as h rightwards arrow 0 of space b subscript n plus cos pi left parenthesis 2 n minus h right parenthesis equals b subscript n plus cos 2 n pi equals b subscript n plus 1
F o r space r i g h t space h a n d space l i m i t comma space l e t space x equals 2 n plus h comma space a s space x rightwards arrow 2 n comma space h rightwards arrow 0
R H L equals limit as h rightwards arrow 0 of space a subscript n plus sin pi left parenthesis 2 n plus h right parenthesis equals a subscript n plus sin 2 n pi equals a subscript n
S i n c e comma space f space i s space c o n t i n u o u s
rightwards double arrow L H L equals f left parenthesis 2 n right parenthesis equals R H L
rightwards double arrow b subscript n plus 1 equals a subscript n equals a subscript n
rightwards double arrow b subscript n plus 1 equals a subscript n
rightwards double arrow a subscript n minus b subscript n equals 1 space.... space left parenthesis i right parenthesis

A t space x equals 2 n plus 1
F o r space l e f t space h a n d space l i m i t comma space l e t space x equals 2 n plus 1 minus h comma space a s space x rightwards arrow 2 n plus 1 comma space h rightwards arrow 0
L H L equals limit as h rightwards arrow 0 of space a subscript n plus sin pi left parenthesis 2 n plus 1 minus h right parenthesis equals b subscript n plus sin left parenthesis 2 n plus 1 right parenthesis pi equals a subscript n
F o r space r i g h t space h a n d space l i m i t comma space l e t space x equals 2 n plus 1 plus h comma space a s space x rightwards arrow 2 n plus 1 comma space h rightwards arrow 0
R H L equals limit as h rightwards arrow 0 of space b subscript n plus 1 end subscript plus cos pi left parenthesis 2 n plus 1 plus h right parenthesis equals b subscript n plus 1 end subscript plus cos left parenthesis 2 n plus 1 right parenthesis pi equals b subscript n plus 1 end subscript minus 1
S i n c e comma space f space i s space c o n t i n u o u s
rightwards double arrow L H L equals f left parenthesis 2 n plus 1 right parenthesis equals R H L
rightwards double arrow a subscript n equals a subscript n equals b subscript n plus 1 end subscript minus 1
rightwards double arrow a subscript n equals b subscript n plus 1 end subscript minus 1
rightwards double arrow a subscript n minus b subscript n plus 1 end subscript equals negative 1 space.... space left parenthesis i i right parenthesis
F o r m space left parenthesis i right parenthesis comma space o p t i o n space B space i s space c o r r e c t
F r o m space left parenthesis i i right parenthesis comma space o p t i o n space D thin space i s space c o r r e c t.
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