3.4 Rational Functions, Asymptotes & Discontinuities

3.4 Rational Functions, Asymptotes & Discontinuities

NBPTS Exam Focus: Component 1 assesses advanced understanding of rational functions: distinguishing removable discontinuities from vertical asymptotes via factor multiplicity, calculating horizontal and slant asymptotes, analyzing limit behaviors from the left and right, and addressing student misconceptions about asymptote crossings.

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Anatomy of Rational Functions & Natural Domains

A rational function is any function that can be expressed as the quotient of two polynomial functions:

$$R(x) = \frac{P(x)}{Q(x)} = \frac{a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0}{b_d x^d + b_{d-1} x^{d-1} + \dots + b_1 x + b_0} \quad (Q(x) \not\equiv 0)$$

where $n = \deg(P)$ is the degree of the numerator and $d = \deg(Q)$ is the degree of the denominator ($a_n \ne 0, b_d \ne 0$).

The natural domain of $R(x)$ is the set of all real numbers except the real roots of the denominator polynomial:

$$\text{Dom}(R) = \{x \in \mathbb{R} \mid Q(x) \ne 0\}$$

Every real number $c$ where $Q(c) = 0$ represents a point of discontinuity.

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Classification of Discontinuities: Removable Holes vs. Vertical Asymptotes

To classify a discontinuity at $x = c$, factor $P(x)$ and $Q(x)$ completely over $\mathbb{R}$. Express the rational function in terms of the linear factor $(x - c)$:

$$R(x) = \frac{(x - c)^m \cdot p_1(x)}{(x - c)^k \cdot q_1(x)} \quad \text{where} \quad p_1(c) \ne 0, \; q_1(c) \ne 0$$

Here $m \ge 0$ is the multiplicity of root $c$ in the numerator, and $k \ge 1$ is its multiplicity in the denominator.

1. Removable Discontinuity (Hole): $m \ge k$

If the factor $(x - c)$ appears in the numerator with multiplicity greater than or equal to its multiplicity in the denominator, the factor cancels completely from the denominator:

• The function $R(c)$ is undefined ($0/0$ indeterminate form).
• However, the two-sided limit $\lim_{x \to c} R(x)$ exists and is finite: $$L = \lim_{x \to c} R(x) = \lim_{x \to c} \frac{(x - c)^{m-k} p_1(x)}{q_1(x)}$$
• Geometrically, the graph has a removable discontinuity (hole) at the point $(c, L)$.

2. Non-Removable Infinite Discontinuity (Vertical Asymptote): $m < k$

If $k > m$, the factor $(x - c)$ remains in the denominator after full algebraic cancellation:

• As $x \to c$, the denominator approaches $0$ while the numerator approaches a non-zero constant.
• The one-sided limits diverge to $\pm \infty$. Therefore, the line $x = c$ is a vertical asymptote.

Behavior of Branches Near Vertical Asymptotes (Multiplicity Rule)

The net multiplicity in the denominator after cancellation is $r = k - m$:

• Odd Net Multiplicity ($r = 1, 3, 5, \dots$): The expression changes sign across $x = c$. Consequently, the branches on either side of the vertical asymptote diverge in opposite directions: $$\lim_{x \to c^-} R(x) = +\infty \implies \lim_{x \to c^+} R(x) = -\infty \quad \text{(or vice versa)}$$
• Even Net Multiplicity ($r = 2, 4, 6, \dots$): The expression does not change sign across $x = c$. Consequently, the branches diverge in the same direction: $$\lim_{x \to c^-} R(x) = \lim_{x \to c^+} R(x) = +\infty \quad \text{or} \quad \lim_{x \to c^-} R(x) = \lim_{x \to c^+} R(x) = -\infty$$

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End Behavior Asymptotes: Horizontal, Slant & Non-Linear

While vertical asymptotes describe local singularities at finite inputs, end-behavior asymptotes describe the asymptotic trajectory of $R(x)$ as $x \to \infty$ and $x \to -\infty$. End behavior is governed entirely by the ratio of leading terms:

$$\lim_{x \to \pm \infty} R(x) = \lim_{x \to \pm \infty} \frac{a_n x^n}{b_d x^d} = \lim_{x \to \pm \infty} \frac{a_n}{b_d} x^{n - d}$$

Calculating Slant (Oblique) Asymptotes via Polynomial Long Division

When $n = d + 1$, polynomial long division decomposes $R(x)$ into a linear polynomial plus a proper rational remainder:

$$R(x) = \frac{P(x)}{Q(x)} = (mx + b) + \frac{r(x)}{Q(x)} \quad \text{where} \quad \deg(r) < \deg(Q)$$

Because $\deg(r) < \deg(Q)$, the remainder fraction vanishes at infinity:

$$\lim_{x \to \pm \infty} [R(x) - (mx + b)] = \lim_{x \to \pm \infty} \frac{r(x)}{Q(x)} = 0$$

Therefore, the straight line $y = mx + b$ is the slant asymptote.

Worked Division Example: Find the slant asymptote of $R(x) = \frac{2x^3 - 8x}{x^2 - x - 2}$.

1. Factoring: $R(x) = \frac{2x(x - 2)(x + 2)}{(x - 2)(x + 1)}$. For $x \ne 2$, $R(x) = \frac{2x^2 + 4x}{x + 1}$.
2. Performing polynomial division of $(2x^2 + 4x)$ by $(x + 1)$: $$\frac{2x^2 + 4x}{x + 1} = 2x + 2 - \frac{2}{x + 1}$$
3. As $x \to \pm \infty$, $\frac{-2}{x + 1} \to 0$. Hence the slant asymptote is $y = 2x + 2$.

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Crossing Asymptotes & The 7-Step Rational Graphing Algorithm

The Fundamental Truth: Can a Graph Cross an Asymptote?

• Vertical Asymptotes: A graph can never cross a vertical asymptote $x = c$. By definition, $c$ is not in the domain of $R$; evaluating $R(c)$ yields a division by zero.
• Horizontal & Slant Asymptotes: A graph can and frequently does cross horizontal and slant asymptotes! Horizontal and slant asymptotes describe behavior only in the infinite tails ($x \to \pm \infty$). They place no restriction whatsoever on local behavior for finite $x$.
• Determining Intersection Points: To determine whether and where $R(x)$ crosses its horizontal asymptote $y = L$ (or slant asymptote $y = mx + b$), set $R(x) = y_{\text{asymptote}}$ and solve for real roots $x$.

Worked Intersection Example:

$$R(x) = \frac{2x^2 + 5x - 3}{x^2 + 1}$$

1. Degrees are equal ($n = 2, d = 2$). Leading coefficients are $2$ and $1$. Horizontal asymptote is $y = \frac{2}{1} = 2$.
2. Set $R(x) = 2$ to test for crossings: $$\frac{2x^2 + 5x - 3}{x^2 + 1} = 2 \implies 2x^2 + 5x - 3 = 2(x^2 + 1) = 2x^2 + 2$$ $$5x - 3 = 2 \implies 5x = 5 \implies x = 1$$
3. The graph crosses its horizontal asymptote at the exact coordinate $(1, 2)$!

The 7-Step Algorithm for Graphing Rational Functions

1. Factor Completely: Factor numerator and denominator over $\mathbb{R}$ to determine the domain.
2. Identify Removable Discontinuities: Cancel matching factors. For each cancelled factor $(x - c)$, compute $L = \lim_{x \to c} R(x)$ using the simplified form, and plot an open circle at $(c, L)$.
3. Identify Vertical Asymptotes: Find roots of the remaining denominator. Note branch behavior using the multiplicity rule.
4. Identify End-Behavior Asymptote: Compare degrees to determine horizontal or slant asymptote.
5. Find Intercepts:
   • $y$-intercept: compute $R(0)$ (if $0 \in \text{Dom}(R)$).
   • $x$-intercepts: set remaining numerator to zero (roots where $P(x) = 0$).
6. Check for Asymptote Crossings: Solve $R(x) = y_{\text{asymptote}}$. Plot any intersection points.
7. Sign Analysis & Sketch: Partition the real line using all $x$-intercepts and vertical asymptotes. Construct a sign chart to determine whether $R(x)$ is above ($+$) or below ($-$) the $x$-axis on each sub-interval, then sketch smooth curves.

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Pedagogical Misconceptions & Secondary Student Traps (NBPTS Focus)

1. "A Graph Can Never Cross an Asymptote"

• Student Manifestation: When graphing $R(x) = \frac{2x^2 + 5x - 3}{x^2 + 1}$, students refuse to draw the curve through $(1, 2)$, bending the graph unnaturally away from $y = 2$.
• Root Cause: In middle school and Algebra I, asymptotes are often colloquially defined as "a line that a curve gets closer and closer to but never touches or crosses." This informal description is overgeneralized from the reciprocal parent function $y = 1/x$.
• Instructional Remedy: Distinguish between local domain exclusions (vertical asymptotes where $x$ cannot be evaluated) and global asymptotic limits at infinity. Define horizontal asymptotes rigorously using limit notation: $\lim_{x \to \infty} R(x) = L$ specifies the limit at infinity, which says nothing about finite inputs. Show students damped oscillating functions like $f(x) = \frac{\sin(x)}{x}$, which crosses its horizontal asymptote $y = 0$ infinitely many times.

2. Conflating Simplified Fractions with Original Functions

• Student Manifestation: When simplifying $f(x) = \frac{x^2 - 9}{x - 3} = x + 3$, students assert that $f(x)$ is identical to the linear function $g(x) = x + 3$ across all of $\mathbb{R}$, concluding that $f(3) = 6$.
• Instructional Remedy: Emphasize the formal definition of function equality: two functions $f$ and $g$ are equal if and only if $\text{Dom}(f) = \text{Dom}(g)$ and $f(x) = g(x)$ for all $x$ in that common domain. Since $3 \notin \text{Dom}(f)$, $f \ne g$. Graph both functions side-by-side using dynamic geometry software, demonstrating that $f$ has a missing point (hole) at $(3, 6)$.

3. Blind Rule Application for Slant Asymptotes

• Student Manifestation: Students attempt to find a slant asymptote when $\deg(P) = \deg(Q) + 2$ by dividing and keeping only the linear part, or assume any degree mismatch produces a slant asymptote.
• Instructional Remedy: Require students to always state the degree comparison condition $\deg(P) = \deg(Q) + 1$ before initiating polynomial division. Show that if $\deg(P) = \deg(Q) + 2$, the quotient is parabolic ($y = ax^2 + bx + c$), creating a parabolic asymptote rather than a slant line.