Point of discontinuity? Pre-calc?

Question

I need help determining when a domain restriction is a point of discontinuity or an asymptote.

For example, the function (X)/(X2-5X)

X divided by X squared - five X

The asymptote is at X=5 and the point is at 0. How can you find these things out without knowing the shape of the graph etc.

I want to know how to do this for all functions not just this one.

Thanks

Answer 1

In a rational function, an asymptote (vertical) will occur at x = a when the denominator but not the numerator is divisible by x - a.  On the other hand, a point of discontinuity will occur at x = a when both the numerator and denominator are divisible by x - a.  In this case,

f(x) = x/(x^2 - 5x) = x/[(x)(x - 5)]

x - 5 divides the denominator but not the numerator, so you have a vertical asymptote at x = 5.  However, x = x - 0 divides both the numerator and denominator of f(x), so there is a point of discontinuity at x = 0.