identify any points of discontinuity and determine if they are removable, infinite, or jump discontinuities

Question

f(x)=[sin(x^2-4)]^-1 

 

give reasons on why it is the specific discontinuity

Answer 1

Well because the sine function is raised to the power of -1, the function can be written as:

 

 f(x) = 1/ ( sin(x^2-4))

 

This function has an infinite number of discontinuities because the sine function is zero at

infinitely many points, specifically 0,pi,2*pi, 3*pi, ....,  k*pi, for integer k

 

Specifically, the discontinuities will occur when (x+2)(x-2) = k*pi

 

                        x^2 - 4 - k*pi = 0

                      x^2 - (k*pi + 4) = 0

 

                      x^2 = (k*pi +4) 

 

                     x = +or- (k*pi +4)