Rational Functions and Vertical Asymptotes
Introduced| English | Meaning |
|---|---|
| denominator/dɪˈnɒmɪneɪtə/ | The lower part of a fraction, indicating the divisor or the number of equal parts in a whole. |
| vertical asymptote/ˈvɜːtɪkl ˈæsɪmptəʊt/ | A vertical line approached by a curve as its output grows without bound near a particular input. |
| asymptote/ˈæsɪmptəʊt/ | A line that a curve approaches arbitrarily closely as its input or output tends to a particular limit. |
| numerator/ˈnjuːməreɪtə/ | The upper part of a fraction, showing how many of the denominator's equal parts are represented. |
| multiplicity/ˌmʌltɪˈplɪsɪti/ | The number of times a root occurs as a repeated factor of a polynomial. |
Dividing by almost nothing
- Try $1 \div 0.1 = 10$, then $1 \div 0.01 = 100$, then $1 \div 0.001 = 1000$.
- As the bottom shrinks toward zero, the answer explodes toward infinity.
- A rational function does exactly this near an input that zeroes its denominator.
- The graph rockets up or down along an invisible vertical line.
Where the denominator hits zero
- A vertical asymptote 竖直渐近线 is a vertical line the graph races along but never crosses.
- It sits at each input that makes the denominator 分母 zero.
- There the function is undefined, and nearby values grow without limit.
- So to find vertical asymptotes, set the bottom equal to zero and solve.
A vertical asymptote of a rational function occurs where…
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Dividing by a number heading to $0$ makes the value blow up — so a denominator zero (not cancelled) gives a vertical asymptote.
…unless the factor cancels
- If a value zeroes the numerator 分子 too, the shared factor cancels.
- Then that input is a hole, not an asymptote.
- So a vertical asymptote needs a denominator zero that survives the cancelling.
- Always simplify first, then read the leftover denominator zeros.
Drag the vertical asymptote left and right
y = a/(x − b) + c
Change b and the vertical break moves to x = b — the exact input that makes the denominator zero.
If a value makes both the numerator and denominator zero, there may be a hole instead of a vertical asymptote.
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A shared factor cancels, so that input gives a hole (removable). A vertical asymptote needs the denominator zero to survive.
Behavior on each side
- An asymptote 渐近线 describes a limiting line; here the limits are infinite.
- Just left of the line, $f(x) \to +\infty$ or $-\infty$; just right, the same choice again.
- Written with limits: $\lim_{x \to 2^-} f(x) = -\infty$ and $\lim_{x \to 2^+} f(x) = +\infty$, for example.
- Checking a value very close on each side tells you which way each branch goes.
Near a vertical asymptote, the size of $f(x)$ grows without bound toward ____.
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On each side $f(x) \to +\infty$ or $-\infty$; the curve races up or down along the asymptote.
Multiplicity sets the pattern
- The multiplicity 重数 of the denominator zero decides the two-sided pattern.
- Odd multiplicity → the sign flips across the line: one side $+\infty$, the other $-\infty$.
- Even multiplicity → the sign is the same: both sides go the same way.
- So $\frac{1}{x-2}$ (multiplicity 1) diverges oppositely, while $\frac{1}{(x-2)^2}$ goes up on both sides.
If a denominator zero has odd multiplicity, the two sides of the asymptote…
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Odd multiplicity flips the sign as you cross, so the two sides diverge oppositely. Even multiplicity sends both sides the same way.
Select all true statements about vertical asymptotes.
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A graph never crosses its vertical asymptote — it only races alongside it. The other three are correct.
A denominator zero is only a vertical asymptote if it does not cancel with the numerator. Check for common factors first — otherwise you will draw an asymptote where there is really just a hole.
Find the vertical asymptotes of $f(x) = \dfrac{x+1}{(x-3)(x+1)}$.
- Denominator zeros: $x = 3$ and $x = -1$.
- But $x = -1$ also zeroes the numerator, so that factor cancels — a hole, not an asymptote.
- Only $x = 3$ is a vertical asymptote.
A vertical asymptote sits at each surviving zero of the denominator — where the graph shoots toward $\pm\infty$. If a factor cancels with the numerator, that input is a hole instead. The multiplicity of the denominator zero decides whether the two sides diverge the same way or oppositely.
Write from two vertical asymptotes
- The denominator is (x minus 2)(x minus 5), and neither factor cancels.
- Say how many vertical asymptotes there are.
A rational function has denominator (x minus 2)(x minus 5), and neither factor cancels with the numerator. How many vertical asymptotes does it have?
Two uncancelled denominator zeros give two asymptotes.
How many vertical asymptotes are there?
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Each surviving denominator zero is its own asymptote.