where does the function f(x)=(2x^2-7x-15)/(x^2-x-20)
have
a) an essential discontinuity; and
b) removable discontinuity?
$\displaystyle \frac{2x^2-7x-15}{x^2-x-20} = \frac{(2x+3)(x-5)}{(x+4)(x-5)}$
note that the factor (x-5) in the numerator and denominator will divide out ... f(x) has a removable discontinuity, also known as a "hole", at x = 5.
f(x) has a vertical asymptote, a non-removable discontinuity, at x = -4.