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

$b=1$