Examining Discontinuity

[Jason76]

Given:

\(\displaystyle f(x) = \dfrac{x + 3}{x^{3} - 2x^{2} - 15x}\) IF x does NOT equal \(\displaystyle 7\)

Otherwise \(\displaystyle f(x) = 11\)

Factoring Out:

\(\displaystyle \dfrac{x + 3}{x(x^{2} - 2x - 15)}\)

\(\displaystyle \dfrac{x + 3}{x(x - 5)(x + 3)}\)

Answer

I see how these came to be, as far as being discontinuities (discontinuities are found in the denominator by setting each linear factor to \(\displaystyle 0\) and solving for \(\displaystyle x\)). But some hints on how we can tell they are distinct types of discontinuity.

removable discontinuity at \(\displaystyle x = -3\)

asymptotic discontinuity at \(\displaystyle x = 0\)

asymptotic discontinuity at \(\displaystyle x = 5\)

point discontinuity at \(\displaystyle x = 7\)

 

[daon2]

Removable discontinuities (for rational functions) can be "cancelled" algebraically. If it cannot be cancelled then it is non-removable, or "asymptotic" according to your source. Think of a removable discontinuity as one that does not affect the graph, as the hole is infinitely tiny. The point discontinuity is there because the limit as x approaches 7 exists, but does not equal the function's value.

 

[Jason76]

ok makes sense, thanks.