Question

Determine if the given function has any points of discontinuity. Explain your reasoning.



There is a point of discontinuity at x = b because the denominator has the factor x – b.
There are points of discontinuity at both x = – b and x = b because the numerator has factors of x + b and x – b.
There is a point of discontinuity at x = -b only because the factor of x – b is common to both the numerator and denominator.
There is a point of discontinuity at x = b only because the factor of x – b is common to both the numerator and denominator and factors out.

Answer 1

Correct choice: 4.

Reason: the function is
f(x) = (x^2 − b^2)/(x − b) = ((x − b)(x + b))/(x − b).
For x ≠ b the factor (x − b) cancels and f(x) = x + b. The denominator is zero only at x = b, so the original expression is undefined at x = b; after cancellation the limit as x → b equals 2b, so x = b is a removable discontinuity (a hole). There is no discontinuity at x = −b because the denominator there is nonzero (the numerator is zero at x = −b, which just gives the value 0, not a discontinuity). (If b = 0 the same conclusion holds: a removable hole at x = 0.)