Mathematics
Mathematics
Problem
Let . Determine all -values where is discontinuous and classify each discontinuity as removable or non-removable (vertical asymptote).Intuitive explanation
The key is recognizing that discontinuities occur where the denominator equals zero, but you must first check whether the numerator also vanishes there. If both top and bottom share a factor, that's a removable discontinuity (a "hole")—you can cancel and fill it in. If only the denominator is zero, you have a vertical asymptote, which is non-removable and unbridgeable. Factor carefully and compare.
Intuitive explanation
The key is recognizing that discontinuities occur where the denominator equals zero, but you must first check whether the numerator also vanishes there. If both top and bottom share a factor, that's a removable discontinuity (a "hole")—you can cancel and fill it in. If only the denominator is zero, you have a vertical asymptote, which is non-removable and unbridgeable. Factor carefully and compare.