Question 547530
{{{f(x)=(2x^2-x-10)/(2x-5)}}} is a rational function (meaning a function involving at most polynomials and maybe quotients of polynomials). If you thought you would not have to factor and divide polynomials ever again, you were wrong.
With rational functions, you have to factor, or divide often. It is essential figure out what happens when a denominator is zero. For values of x that make a denominator zero, the function is undefined. It could be a hole, or it could be a vertical asymptote.
Factoring, we find:
{{{f(x)=(2x^2-x-10)/(2x-5)=(2x-5)(x+2)/(2x-5)}}}
{{{2x-5=0}}} <---> {{{x=5/2}}} and the function does not exist.
For any other x, {{{f(x)=x+2}}}
So {{{x=5/2}}} is a hole in the line {{{y=x+2}}}.
No vertical asymptote. Just a hole.