• "point of discontinuity" exists when the numerator and denominator have a factor in common.

i.e. $\left( x \right) = \frac{{ - \left( {3x - 8} \right)\left( {x + 5} \right)\left( {2x - 7} \right)}}{{\left( {x + 5} \right)\left( {4x + 9} \right)\left( {3x + 8} \right)\left( {2x - 7} \right)}}$ ; points of discontinuity exist at $x = - 5$ and $x = \frac{7}{2}$ .

• To determine the coordinates of the point of discontinuity:

1) Factor both the numerator and denominator.

2) Simplify the rational expression by cancelling the common factors.

3) Substitute the non-permissible values of x into the simplified rational expression to obtain the corresponding values for the y-coordinate.

i.e. $\left( x \right) = \frac{{ - \left( {3x - 8} \right)\left( {x + 5} \right)\left( {2x - 7} \right)}}{{\left( {x + 5} \right)\left( {4x + 9} \right)\left( {3x + 8} \right)\left( {2x - 7} \right)}}$ ; points of discontinuity exist at $x = - 5$ and $x = \frac{7}{2}$ .

• To determine the coordinates of the point of discontinuity:

1) Factor both the numerator and denominator.

2) Simplify the rational expression by cancelling the common factors.

3) Substitute the non-permissible values of x into the simplified rational expression to obtain the corresponding values for the y-coordinate.