POINT OF DISCONTINUITY

Point Of Discontinuity

Definition Of Point Of Discontinuity

A function is said to have a point of discontinuity at x = a or the graph of the function has a hole at x = a, if the original function is undefined for x = a, whereas the related rational expression of the function in simplest form is defined for x = a.

Video Examples: Finding Discontinuities of Rational Functions

Example of Point of Discontinuity

Consider a function example of Point of Discontinuity .
This function is undefined for x = 2. But the simplified rational expression of this
function, x + 3 which is obtained by canceling (x - 2) both in the numerator and the denominator is
defined at x = 2. Thus we can say that the function f(x) has a point of discontinuity at x = 2.

Solved Example on Point of Discontinuity

Ques: Which of the following would replace the blank so that the rational function will have a point discontinuity?

Choices:

A. x - 11
B. 2x + 13
C. Either A or B
D. x + 1
Correct Answer: C

Solution:

Step 1: The function example of Point of Discontinuity will have a point discontinuity if the denominator contains either of the binomials (x - 11) or (2x + 13).

Quick Summary

Point discontinuities occur when a factor cancels from both the numerator and denominator of a rational function.
These discontinuities are also known as removable discontinuities.
The graph of the function has a 'hole' at the x-value where the discontinuity occurs.

\[ Let f(x) = \frac{g(x)}{h(x)}. If h(a) = 0 but \lim_{x \to a} f(x) exists, then f(x) has a point of discontinuity at x = a. \]

🍎 Teacher Insights

Emphasize the simplification of rational functions. Use graphing calculators to visualize the 'holes' in the graph at the point of discontinuity. Present both algebraic and graphical approaches.

🎓 Prerequisites

Limits
Rational Functions
Factoring Polynomials

Check Your Knowledge

Q1: Which of the following functions has a point discontinuity?

f(x) = \frac{x}{x^2 + 1} f(x) = \frac{x-2}{x^2 - 4} f(x) = \frac{1}{x} f(x) = x^2 + 3x + 2

Q2: At what x-value does the function f(x) = \frac{(x+1)(x-3)}{x-3} have a point of discontinuity?

-1 3 0 No point of discontinuity

Frequently Asked Questions

Q: How do you find a point of discontinuity?
A: Factor the numerator and denominator of the rational function. If a factor cancels out, then there's a point discontinuity at the x-value that makes that factor equal to zero.

Q: Is a point of discontinuity the same as a vertical asymptote?
A: No. A point of discontinuity (hole) is a removable discontinuity, while a vertical asymptote is a non-removable discontinuity where the function approaches infinity.