Limits Involving Infinity and Asymptotes

Infinite limits and vertical asymptotes

If f(x) grows without bound as x→a from one side, we write lim (x→a⁺) f(x) = ∞ (or −∞). Then x = a is a vertical asymptote.

For a rational function, look for values that make the denominator zero after cancelling common factors.

• f(x) = 1/(x − 2): vertical asymptote x = 2. From the right, f→ +∞; from the left, f→ −∞.
• g(x) = (x² − 4)/(x − 2) = x + 2 (x ≠ 2): no asymptote, only a hole at x = 2.

To find the sign, test a value just to the right or left of a.

Limits at infinity and horizontal asymptotes

lim (x→∞) f(x) = L means f(x) approaches L as x grows. Then y = L is a horizontal asymptote.

For a rational function, compare the degrees of the numerator and denominator:

Degrees	lim (x→±∞)
top smaller than bottom	0
top equal to bottom	ratio of leading coefficients
top larger than bottom	±∞ (no horizontal asymptote)

Example. lim (x→∞) (3x² − 5)/(6x² + x) = 3/6 = 1/2.

A quick method: divide every term by the highest power of x in the denominator.

Growth rates

As x→∞, from slowest to fastest:

ln x < xⁿ (any n > 0) < eˣ

So lim (x→∞) x³/eˣ = 0 and lim (x→∞) ln x / x = 0.

Square roots at infinity

√(x²) = |x|. When x→ −∞, |x| = −x, which can flip a sign.

lim (x→ −∞) √(4x² + 1)/x = −2, while the same limit as x→∞ is 2.

Quick summary

• Vertical asymptote: denominator zero after cancelling.
• Horizontal asymptote: compare degrees, or divide by the highest power.
• Exponentials beat powers, powers beat logarithms.
• Watch the sign of √(x²) for x→ −∞.