Question:

For the function \( f(x)=\frac{x+3}{x^2-3x-10} \),

(a) At which point(s) is \( f(x) \) discontinuous and why?

(b) Classify all of your points of discontinuity as removable or not removable and explain why.

(a) At which point(s) is \( f(x) \) discontinuous and why?

(b) Classify all of your points of discontinuity as removable or not removable and explain why.

Solution:

(a) The function is discontinuous at \( x=,-2,5 \), because those are the zeros of \( x^2-3x-10 \), but not zeros of \( x+3 \) so the function is undefined and has vertical asymptotes at both locations.

(b) Both are not removable because they represent breaks and not holes.

(b) Both are not removable because they represent breaks and not holes.