Asymptotes via Limits

Vertical asymptotes: Where the denominator = 0 (and numerator ≠ 0).

lim(x→a) f(x) = ±∞ → vertical asymptote at x = a.

Horizontal asymptotes: Limits at infinity.

lim(x→∞) f(x) = L → horizontal asymptote at y = L.

Rules for rational functions (degree of numerator vs denominator):

Degree num < degree denom → HA at y = 0

Degrees equal → HA at y = leading coefficients ratio

Degree num > degree denom → no HA (oblique or none)

Removable vs non-removable:

If a factor cancels from top and bottom → removable discontinuity (hole), not a vertical asymptote.

If a factor remains in the denominator → vertical asymptote.

Reference:

Wikipedia: Asymptote

https://en.wikipedia.org/wiki/Asymptote