| English | 의미 |
|---|---|
| denominator/dɪˈnɒmɪneɪtə/ | 분모 |
| vertical asymptote/ˈvɜːtɪkl ˈæsɪmptəʊt/ | 수직 점근선 |
| asymptote/ˈæsɪmptəʊt/ | 점근선 |
| numerator/ˈnjuːməreɪtə/ | 분자 |
| multiplicity/ˌmʌltɪˈplɪsɪti/ | 중근 |
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.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
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.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
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.