1.15 Connecting Limits at Infinity and Horizontal Asymptotes

Limits at infinity describe end behavior

A limit at infinity describes what happens to f(x) as x → ∞ or x → -∞. If lim(x→∞) f(x) = L (a finite number), then the line y = L is a horizontal asymptote. This connects a limit statement directly to a graphical feature of the function.

Rational functions: compare degrees

For rational functions, compare the degree of the numerator (n) and denominator (m). If n < m, the limit is 0. If n = m, the limit is the ratio of leading coefficients. If n > m, the limit is ±∞ (no horizontal asymptote, but possibly an oblique/slant asymptote).

Other end-behavior models

For functions with exponentials, terms like e^(-x) → 0 as x → ∞, while e^x → ∞. For radicals, factor out the highest power of x (careful with signs as x → -∞). A function can have two different horizontal asymptotes — one for x → ∞ and another for x → -∞.

Key terms

Horizontal asymptote (definition)

A line y = L that the graph of f approaches as x → ∞ or x → -∞; occurs when lim(x→±∞) f(x) = L.

Rational function: degree numerator < degree denominator

Limit at infinity = 0.

Rational function: degree numerator = degree denominator

Limit at infinity = ratio of leading coefficients.

Rational function: degree numerator > degree denominator

Limit at infinity is ±∞; no horizontal asymptote (possible oblique asymptote).

lim(x→∞) e^(-x)

0 — exponential decay terms vanish as x increases without bound.

Try an AP-style question

lim(x→∞) (3x² + 1)/(5x² − x) = ?

1. 3/5
2. 3
3. 0
4. ∞