Question

What is a horizontal asymptote

Answer 1

A horizontal asymptote is a horizontal line y = L that the graph of a function approaches as x goes to infinity or negative infinity.

Formally:
- y = L is a horizontal asymptote to the right if lim_{x→∞} f(x) = L.
- y = L is a horizontal asymptote to the left if lim_{x→−∞} f(x) = L.

(Those two limits can be different, so a function can have different left and right horizontal asymptotes.)

Intuition: as you go far out to the right or left along the x-axis, the function's values get arbitrarily close to L.

How to find one (common cases):
- For a rational function f(x) = p(x)/q(x) with p, q polynomials:
- If deg(p) < deg(q), the horizontal asymptote is y = 0.
- If deg(p) = deg(q), the horizontal asymptote is y = (leading coefficient of p)/(leading coefficient of q).
- If deg(p) = deg(q) + 1, there is no horizontal asymptote (there may be an oblique/slant asymptote found by division).
- If deg(p) > deg(q) + 1, there is generally no horizontal asymptote.
- For other functions, compute the limits as x→±∞; if the limit exists and equals L, then y = L is a horizontal asymptote.

Examples:
- f(x) = x/(x^2+1) → lim_{x→±∞} = 0, so y = 0 is a horizontal asymptote.
- g(x) = (2x^2+3)/(x^2−1) → lim_{x→±∞} = 2, so y = 2 is a horizontal asymptote.
- h(x) = arctan(x) → lim_{x→∞} = π/2, lim_{x→−∞} = −π/2, so two horizontal asymptotes y = π/2 and y = −π/2.

Note: a graph can cross its horizontal asymptote finitely or infinitely many times and still have that asymptote.