1. The Idea of a Limit at Infinity
A limit at infinity describes the end behavior of a function. It answers the question: "What value does \( f(x) \) approach as \( x \) becomes extremely large (either positive or negative)?"
This is expressed in notation as:
\[ \lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L \]
If the function approaches a horizontal line, that line is a horizontal asymptote.
2. Horizontal Asymptotes
A horizontal asymptote is a horizontal line \( y = L \) that a function approaches as \( x \to \infty \) or \( x \to -\infty \).
3. Finding Limits at Infinity of Rational Functions
For rational functions, compare the degrees of numerator and denominator.
Rule 1: Numerator degree < Denominator degree → limit = 0
\[ \lim_{x \to \infty} \frac{2x + 1}{x^2 + 5} = 0 \]
Rule 2: Numerator degree = Denominator degree → limit = ratio of leading coefficients
\[ \lim_{x \to \infty} \frac{3x^2 - x}{x^2 + 4x} = 3 \]
Rule 3: Numerator degree > Denominator degree → limit does not exist (∞ or −∞)
\[ \lim_{x \to \infty} \frac{x^3 + 2}{x^2 - 1} = \infty \]
Check Yourself
Check Yourself 1: \(\lim_{x \to \infty} \frac{6x^2 - 3x + 1}{2x^2 + 5x - 4}\)
Degrees equal → ratio of leading coefficients:
\[ \frac{6}{2} = 3 \]
Check Yourself 2: \(\lim_{x \to -\infty} \frac{4x}{x^3 + 1}\)
Numerator degree < denominator degree → limit = 0
Check Yourself 3: \(\lim_{x \to \infty} \frac{2x^2 + 1}{3x^2 - 5}\)
Solution: Divide by x² … → Limit = 2/3.
Check Yourself 4: \(\lim_{x \to \infty} \frac{4x - 7}{2x + 3}\)
Solution: Divide by x … → Limit = 2.
Check Yourself 5: \(\lim_{x \to \infty} \frac{-5x^2 + 2}{2x^2 + 7}\)
Solution: Divide by x² … → Limit = –5/2.