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Find the value of a and b such that the function is continuous at x=0. f(x) =  x+sin x/ sin((a+1)x)     if -π0  
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Asked by saikatsukai514 | 22 Apr, 2020, 21:49: PM
answered-by-expert Expert Answer
f(x) is continuous at x = 0
Therefore, LHL = RHL = f(0)
LHL = f(0)
rightwards double arrow limit as x rightwards arrow 0 to the power of minus of f left parenthesis x right parenthesis equals f left parenthesis 0 right parenthesis rightwards double arrow limit as x rightwards arrow 0 of fraction numerator x plus sin x over denominator sin open square brackets open parentheses a plus 1 close parentheses x close square brackets end fraction equals 2 rightwards double arrow limit as x rightwards arrow 0 of space open parentheses fraction numerator x over denominator sin open square brackets open parentheses a plus 1 close parentheses x close square brackets end fraction plus fraction numerator sin x over denominator sin open square brackets open parentheses a plus 1 close parentheses x close square brackets end fraction close parentheses equals 2 rightwards double arrow limit as x rightwards arrow 0 of space open parentheses fraction numerator left parenthesis a plus 1 right parenthesis x over denominator sin open square brackets open parentheses a plus 1 close parentheses x close square brackets end fraction cross times fraction numerator 1 over denominator left parenthesis a plus 1 right parenthesis end fraction plus fraction numerator sin x over denominator x end fraction cross times fraction numerator left parenthesis a plus 1 right parenthesis x over denominator sin open square brackets open parentheses a plus 1 close parentheses x close square brackets end fraction cross times fraction numerator 1 over denominator left parenthesis a plus 1 right parenthesis end fraction close parentheses equals 2 A s space x rightwards arrow 0 comma space left parenthesis a plus 1 right parenthesis x rightwards arrow 0 rightwards double arrow fraction numerator 1 over denominator left parenthesis a plus 1 right parenthesis end fraction limit as x rightwards arrow 0 of space open parentheses fraction numerator left parenthesis a plus 1 right parenthesis x over denominator sin open square brackets open parentheses a plus 1 close parentheses x close square brackets end fraction close parentheses plus fraction numerator 1 over denominator left parenthesis a plus 1 right parenthesis end fraction limit as x rightwards arrow 0 of fraction numerator sin x over denominator x end fraction cross times limit as x rightwards arrow 0 of fraction numerator left parenthesis a plus 1 right parenthesis x over denominator sin open square brackets open parentheses a plus 1 close parentheses x close square brackets end fraction equals 2 rightwards double arrow fraction numerator 1 over denominator a plus 1 end fraction plus fraction numerator 1 over denominator a plus 1 end fraction equals 2 rightwards double arrow fraction numerator 2 over denominator a plus 1 end fraction equals 2 rightwards double arrow 2 a plus 2 equals 2 rightwards double arrow a equals 0 space... space left parenthesis i right parenthesis R H L equals limit as x rightwards arrow 0 to the power of plus of f left parenthesis x right parenthesis equals limit as x rightwards arrow 0 of fraction numerator 2 open parentheses e to the power of sin space b x end exponent minus 1 close parentheses over denominator b x end fraction equals 2 equals limit as x rightwards arrow 0 of fraction numerator 2 open parentheses e to the power of sin space b x end exponent minus 1 close parentheses over denominator b x end fraction cross times fraction numerator sin space b x over denominator sin space b x end fraction equals limit as x rightwards arrow 0 of fraction numerator 2 open parentheses e to the power of sin space b x end exponent minus 1 close parentheses over denominator sin space b x end fraction cross times fraction numerator sin space b x over denominator b x end fraction equals 2 open square brackets limit as x rightwards arrow 0 of fraction numerator open parentheses e to the power of sin space b x end exponent minus 1 close parentheses over denominator sin space b x end fraction cross times limit as x rightwards arrow 0 of fraction numerator sin space b x over denominator b x end fraction close square brackets equals 2 f left parenthesis 0 right parenthesis equals 2 b space c a n space t a k e space a n y space r e a l space v a l u e space o t h e r space t h a n space 0.
Answered by Renu Varma | 23 Apr, 2020, 11:28: AM

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