Please answer as fast as possible 

Asked by jain.pradeep | 26th Sep, 2019, 11:18: 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 x plus sin x over denominator sin left parenthesis a plus 1 right parenthesis x end fraction
equals limit as x rightwards arrow 0 of fraction numerator 1 plus begin display style fraction numerator sin x over denominator x end fraction end style over denominator begin display style fraction numerator sin left parenthesis a plus 1 right parenthesis x over denominator x end fraction end style end fraction
equals limit as x rightwards arrow 0 of space fraction numerator 1 plus begin display style fraction numerator sin x over denominator x end fraction end style over denominator begin display style fraction numerator sin left parenthesis a plus 1 right parenthesis x over denominator left parenthesis a plus 1 right parenthesis x end fraction end style end fraction cross times fraction numerator 1 over denominator a plus 1 end fraction
equals fraction numerator 1 plus 1 over denominator 1 end fraction cross times fraction numerator 1 over denominator a plus 1 end fraction
equals fraction numerator 2 over denominator a plus 1 end fraction
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 2 fraction numerator e to the power of sin b x end exponent minus 1 over denominator b x end fraction
equals limit as x rightwards arrow 0 of 2 fraction numerator e to the power of sin b x end exponent minus 1 over denominator b x end fraction cross times fraction numerator sin b x over denominator sin b x end fraction
equals limit as x rightwards arrow 0 of 2 fraction numerator e to the power of sin b x end exponent minus 1 over denominator sin b x end fraction cross times fraction numerator sin b x over denominator b x end fraction
equals 2
F o r space f space t o space b e space c o n t i n u o u s comma space w e space m u s t space h a v e
f left parenthesis 0 right parenthesis equals L H L equals R H L
fraction numerator 2 over denominator a plus 1 end fraction equals 2 rightwards double arrow a equals 0
b space m a y space b e space a n y space r e a l space n u m b e r space o t h e r space t h a n space 0.

Answered by Renu Varma | 27th Sep, 2019, 10:40: AM