Question
Tue June 19, 2012 By: Shubhanshu Yadav

Make the function continuous.

Expert Reply
Thu June 28, 2012
Answer : Given : f(x)= (4– 1)3/ ( sin x/4 * log  (1 + (x2/3) ) )
                 to find : f(0) such that f(x) becomes continuous at x=0
 
            for f(x) to be continuous at x=0, 
                       limx->0 f(x) = f(0)       condition should hold true.

=>   f(0) = limx->0 f(x) =   limx->0 (4– 1)3/ ( sin x/4 * log  (1 + (x2/3) ) )
 
=> f(0) =( (4– 1) /x)/  ( ( sin (x/4) )/(4*(x/4)) * ((log  (1 + (x2/3) )) /(x2/3))
                                     { here we divided by x3 in numerator and denomenator and                                           adjusted the values to get standard forms }
 
here we will use  limx->0 (ax -1) /x = log a 
                          limx->0 (sin x )/x =1
                          limx->0  ( log(1+x) )/x =1
we get ,
=>f(0) = (log e4)3 / (1/4 ) * (1/3)   = 12 (loge 4)3 Answer 
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