Piece wise function sin(x-a)/(x-3), x not equal to 3 and other function equal b where x equals 3 . for what values of a and b function is continuous?

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Answer 1 · 2018-07-07T01:42:59

#a= 3 +n pi,qquad n in ZZ#
#b=1#

The given function is

#f(x) = sin(x-a)/(x-3) qquad "for " x ne 3,qquadqquad f(3)=b#

Now, for the function to be continuous, the limit of #f(x)# at #x=3# must exist, and be equal to #f(3)#.

Since #x-3# vanishes at #x=3#, the numerator, #sin(x-a)# must vanish there for the limit to exist. Thus

#sin(3-a)=0 qquad implies qquad a=3+n pi#

Again

#lim_{x to 3} sin(x-3-n pi)/(x-3) = lim_{h to 0} sin h/h = 1#

so we must have

#b=1#