Question
Sat June 09, 2012 By: Nevin

sir/madam if f(x)defined by the following function is continuous at x = pi find the values of a and b

Expert Reply
Fri June 29, 2012
Answer : Given : 

             f(x) = (sin 3x - 3sinx)/(pi-x),when  x < pi

                  = a                                     ,  when  x =pi

                 = [b((5 + cos x)1/2- 2)]/(pi – x)2 , when x > pi

 to find : the values of a and b such that f(x) is continuous at x= pi

LHL at x= pi

= lim x-> pi- (sin 3x - 3sinx)/(pi-x)3

= lim x-> pi- ((3 sin x - 4sin 3 x ) -3sinx)/(pi-x)3     { applying the formula                                                                     sin3x = 3 sin x - 4sin 3 x }

= lim x-> pi-  - ( 4 sin3x) /(pi-x)3

now let h-> 0 when x-> pi - , h= pi - x

= lim h-> 0 -4 ( sin3 ( pi -h)) / h3 

= lim h-> 0  -4 ( sin (h)) / h3            

= - 4                        { aplying the condition lim x->0 (sin x )/ x =1 }

 

therefore as f(x) is continuous at x = pi , LHL = f( pi) 

      => a= - 4

 

 now RHL at x = pi 

  =lim x -> pi +   [b((5 + cos x)1/2- 2)]/(pi – x)2

 let h -> 0 when x -> pi + ,  x = pi + h

= lim h-> 0  [b((5 + cos (pi +h)1/2- 2)]/(pi –(pi+h))2

= lim h-> 0   [b((5 - cos h)1/2- 2)]/(h)2         { using cos ( pi +x ) = - cos x } 

= lim h-> 0  [ b ( 5- cos h)2 - 22 )  ] / h2 ((5 - cos h)1/2 + 2)  { after                                                                                              rationalising  and using (a+b ) (a-b) = a2- b2 } 

= lim  h- >0 [ b ( 5- cos h - 4 )  ] / h2 ((5 - cos h)1/2 + 2)  

= lim h-> 0 [ b ( 1- cos h )  ] / h2 ((5 - cos h)1/2 + 2)  

= lim  h->0 [ b (  sin2 (h/2) )  ] / [ 2  (h2 /4 )((5 - cos h)1/2 + 2)]

        { using 1- cos 2x = 2 sin2 x  and                                                               dividing by 2 and adjusting                                                                values} 

=  ( b ) /  [2 *4]     { applying the formula lim x->0 (sin x )/x                                                                =1  and applying limit  }

= b /8 

as we know that f(x) is continuous at x= pi , RHL = LHL = f(pi)

=> b = 8 * (-4)

=> b= -32

Answer : a = -4 and b = -32  

Mon September 05, 2016

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