Question

Find the horizontal asymptote of the function .

y =

Answer 1

It looks like the formula didn’t come through — could you paste the full function y = ?

Meanwhile, here are quick rules you can use to find horizontal asymptotes (HA):

1. Definition (general): y = L is a horizontal asymptote if lim_{x→∞} f(x) = L or lim_{x→−∞} f(x) = L. You should check both ±∞ (they can be different).

2. For rational functions f(x) = P(x)/Q(x) (polynomials):
- deg(P) < deg(Q) → HA: y = 0.
- deg(P) = deg(Q) → HA: y = (leading coefficient of P) / (leading coefficient of Q).
- deg(P) = deg(Q) + 1 → no horizontal asymptote; there may be a slant (oblique) asymptote found by polynomial long division.
- deg(P) > deg(Q) + 1 → no horizontal asymptote (end behavior grows without approaching a constant).

3. Other common cases:
- Exponentials: e^{−x} → HA y = 0 as x→∞; e^{x} has no HA as x→∞.
- Inverse trig: arctan x → HA y = ±π/2 as x→±∞.
- Use limits if unsure.

Examples:
- y = (3x^2 + 5)/(2x^2 − 1) → deg equal → HA y = 3/2.
- y = (2x + 1)/(x^2 + 1) → deg numerator < denominator → HA y = 0.
- y = (x^3)/(x^2 + 1) → deg numerator > denominator by 1 → no horizontal asymptote (slant asymptote instead).

If you post the exact function, I’ll compute the horizontal asymptote(s) for you.