An asymptote is a line a curve gets arbitrarily close to but never touches.
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y = 1/(x - a)
Spotting asymptotes of a rational function
- Vertical: where the denominator is zero (and the numerator isn't) — the curve shoots to ±∞ there.
- Horizontal: the value y approaches as x → ±∞; for 1/(x−a) that's y = 0.
- Slant: appears when the numerator's degree is exactly one more than the denominator's.
Your turn
Where is the vertical asymptote of y = 1/(x − 3)?
A curve can cross a horizontal or slant asymptote — 'never touches' is only guaranteed for vertical asymptotes. The asymptote describes long-run behaviour, not a fence.
Three types
- Vertical: where the function blows up (e.g. x = 0 for 1/x).
- Horizontal: what y approaches as x → ±∞.
- Slant (oblique): a slanted line approached at infinity.