Unit 2 · Topic 1.10
Rational Functions and Holes
Use common-factor multiplicities and finite limits to classify removable discontinuities and determine the exact coordinates of missing points.
1. What Is a Hole?
A hole is a single missing point on a graph. The function is undefined at the hole's input, but nearby outputs approach one finite value.
\[\lim_{x\to c}R(x)=L\quad\text{while}\quad R(c)\text{ is undefined}.\]
The hole is located at \((c,L)\). It is also called a removable discontinuity because defining the missing output as \(L\) would fill the gap.
2. Shared Factors Create Hole Candidates
Suppose \((x-c)^p\) is a numerator factor and \((x-c)^q\) is a denominator factor. The original denominator makes \(x=c\) an excluded input.
\[p\ge q>0\quad\Longrightarrow\quad\text{the graph has a hole at }x=c.\]
If \(q>p\), a denominator factor remains after cancellation and \(x=c\) is a vertical asymptote instead.
3. Multiplicity Classification Table
| Numerator multiplicity \(p\) | Denominator multiplicity \(q\) | After cancellation | Classification |
|---|
| \(p=q\) | Equal | The shared factor disappears | Hole |
| \(p>q\) | Numerator larger | A numerator factor remains | Hole |
\(p| Denominator larger | A denominator factor remains | Vertical asymptote | |
Multiplicity classifies the discontinuity. The simplified rule then supplies the missing y-coordinate.
4. Reliable Hole-Finding Workflow
- Factor the numerator and denominator.
- Record every zero of the original denominator.
- Compare multiplicities at each shared real zero.
- Cancel common factors but preserve all restrictions.
- For each hole input \(c\), evaluate the simplified rule at \(c\).
- Report the full coordinate \((c,L)\) and the limit \(\lim_{x\to c}R(x)=L\).
5. Equal-Multiplicity Example
Locate the hole of
\[A(x)=\frac{x^2-9}{x-3}=\frac{(x-3)(x+3)}{x-3}.\]
The original domain excludes \(x=3\). For every other input, \(A(x)=x+3\). Therefore
\[\lim_{x\to3}A(x)=\lim_{x\to3}(x+3)=6.\]
The graph is the line \(y=x+3\) with the point \((3,6)\) removed.
6. Larger Numerator Multiplicity Example
Analyze
\[B(x)=\frac{(x-2)^2(x+3)}{(x-2)(x+1)}.\]
The original domain excludes \(x=2\) and \(x=-1\). Cancellation gives
\[B(x)=\frac{(x-2)(x+3)}{x+1},\qquad x\ne2,-1.\]
- At \(x=2\), numerator multiplicity 2 exceeds denominator multiplicity 1, so there is a hole.
- The simplified output at 2 is 0, so the hole is \((2,0)\).
- At \(x=-1\), a denominator factor remains, so \(x=-1\) is a vertical asymptote.
The missing point \((2,0)\) is not an x-intercept because \(B(2)\) is undefined.
7. Multiple Holes and an Asymptote
Consider
\[C(x)=\frac{(x+2)^2(x-1)}{(x+2)(x-1)(x-4)}.\]
The original restrictions are \(x=-2,1,4\), and the simplified rule is
\[C(x)=\frac{x+2}{x-4},\qquad x\ne-2,1,4.\]
| Excluded input | Simplified behavior | Classification | Coordinate or line |
|---|
| \(-2\) | \((x+2)/(x-4)\to0\) | Hole | \((-2,0)\) |
| 1 | \((x+2)/(x-4)\to-1\) | Hole | \((1,-1)\) |
| 4 | Denominator remains zero | Vertical asymptote | \(x=4\) |
8. Left, Right, and Two-Sided Limits
At a hole, both sides approach the same finite output:
\[\lim_{x\to c^-}R(x)=\lim_{x\to c^+}R(x)=\lim_{x\to c}R(x)=L.\]
If the one-sided limits differ, there is no single finite hole coordinate. If either side is unbounded, the behavior is associated with a vertical asymptote instead.
9. Numerical Evidence for a Hole
For \(A(x)=(x^2-9)/(x-3)\), nearby outputs approach 6 even though \(A(3)\) is undefined.
| \(x\) | 2.9 | 2.99 | 3 | 3.01 | 3.1 |
|---|
| \(A(x)\) | 5.9 | 5.99 | Undefined | 6.01 | 6.1 |
|---|
The left- and right-side values close in on the same finite number, confirming the hole at \((3,6)\).
10. Filling the Hole
A removable discontinuity can be repaired by defining the missing output to equal the limit. For the equal-multiplicity example, define
\[G(x)=\begin{cases}\dfrac{x^2-9}{x-3},&x\ne3,\\6,&x=3.\end{cases}\]
Then \(G\) agrees with the line \(x+3\) everywhere and is continuous at \(x=3\). This new definition does not change the original function; it constructs an extended function.
11. Distinguish Nearby Graph Features
| Feature | Function value at \(x=c\) | Nearby output behavior |
|---|
| Hole | Undefined | Approaches one finite value |
| Vertical asymptote | Undefined | Becomes unbounded on at least one side |
| x-intercept | Defined and equals 0 | Graph contains \((c,0)\) |
| Ordinary point | Defined | Graph contains the corresponding point |
12. Technology Workflow
- Factor and preserve original denominator restrictions.
- Classify each restriction using multiplicities.
- Evaluate the simplified rule to find each missing y-coordinate.
- Use a table approaching from both sides to verify the finite limit.
- Graph the original and simplified rules together.
- Mark holes manually if the graphing tool does not display them.
Some graphing tools draw the simplified curve without showing the missing point, so algebraic domain analysis is essential.
13. Contextual Meaning
In a model, a hole means the formula is undefined at one input even though nearby outputs approach a reasonable finite value. Whether filling that value is meaningful depends on the original context and definition.
Do not automatically extend a model across a hole without explaining what the excluded input represents.
14. Common Errors and AP Reasoning
- Reporting only the hole's x-coordinate.
- Substituting into the original undefined expression to find the y-coordinate.
- Canceling a factor and restoring the excluded input.
- Calling every common-factor location a hole without comparing multiplicities.
- Calling a hole on the x-axis an x-intercept or zero.
- Confusing a finite limit with the actual function value.
- Trusting a graphing tool that silently fills the missing point.
Complete response pattern: identify the original restriction, compare multiplicities, simplify with the restriction preserved, evaluate the limiting rule, and report \((c,L)\) with its limit statement.