For the given function, ƒ, determine if the function is discontinuous at a given point. State which condition of continuity fails or state the function is conti

Question

For the given function, ƒ, determine if the function is discontinuous at a given point. State which condition of continuity fails or state the function is continuous at x = a.
f(x) = (x^3-9x)/(x^2+11x+24), a=-3

Answer 1

When simplified by factoring, you get (x(x-3)/(x+8) so there is a HOLE at x = -3 due to cancellation of a factor (x+3) that appeared in both numerator & denominator of this rational function.  There is a vertical asymptote: x = -8. There is also a slant (oblique) asymptote because the (simplified) version of f has degree one more in numerator than in denominator.

The hole is considered to be a removable discontinuity.