1.15 Limits at Infinity & Horizontal Asymptotes

End behavior

lim(x→±∞) f(x) = L means y = L is a horizontal asymptote — the function levels off toward L at the far ends.

Rational shortcut

Compare degrees: if the bottom degree is bigger, the limit is 0; if degrees are equal, the limit is the ratio of leading coefficients; if the top degree is bigger, there is no horizontal asymptote (it grows). Example: lim(x→∞) (2x+1)/(x−5) = 2.

Key terms

Limit at infinity

The value f approaches as x → ±∞.

Horizontal asymptote

y = L when lim(x→±∞) f(x) = L.

Bottom-heavy

Denominator degree bigger → limit 0.

Equal degrees

Limit = ratio of leading coefficients.

Top-heavy

Numerator degree bigger → no horizontal asymptote.

Try an AP-style question

lim(x→∞) (3x² + 1)/(5x² − x) = ?

1. ∞
2. 0
3. 3/5
4. 3