00:01
This problem gives us the rational function, x squared minus 4 over x cubed minus x squared minus 2x, and it gives us some statements about the discontinuities of the function that we need to determine which one is true.
00:11
So when we do this, let's analyze the function first, and to start analyzing the function, we're going to factor.
00:17
Our x squared minus 4 is a difference of squares that factors to x plus 2 times x minus 2.
00:23
And for our bottom expression, we would need to factor out an x first, which would give us x squared minus x, and then we'd ask ourselves what two numbers multiply to be negative 2 and add to be the negative 1.
00:35
And with the x still out in front, that would factor to x minus 2 and x plus 1.
00:42
Again, with negative 2 and positive 1 multiplying to give you negative 2 and then adding to give you negative 1.
00:47
So when we look at our discontinuities, discontinuities come from shared solutions of factors on the numerator and denominator.
00:55
So since we have a shared factor of x minus 2 and x minus 2, that means the x solution of 2 will no longer be considered as 0, which is what the factor on top would tell you.
01:09
And it's no longer considered a vertical asymptote, which is what the factor on bottom would tell you.
01:13
So again, since they're shared, that means that we have a discontinuity at the x value of 2...