Limits at Infinity & Horizontal Asymptotes (1.5)

Overview

This lecture covers limits at infinity, including key techniques for evaluating them and common types of functions, with an emphasis on rational, radical, and transcendental functions.

Limits at Infinity: Basics

• The limit as x approaches infinity of 1/x is 0; large denominators yield small values close to zero.
• For x approaches negative infinity, 1/x still approaches 0 but from the negative side.
• The graph of 1/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.

Vertical & Horizontal Asymptotes

• Vertical asymptote: occurs where the function approaches infinity, such as x = 0 for 1/x.
• Horizontal asymptote: represents the value y approaches as x goes to ±infinity, like y = 0 for 1/x and 1/x².

General Theorems & Techniques

• For any positive exponent r, lim(x→∞) [1/x^r] = 0.
• If the degree of the denominator is greater than the numerator ("bottom-heavy" rational function), the limit as x→∞ is 0.
• When degrees are equal in numerator and denominator, divide leading coefficients to get the limit.
• Use algebraic manipulation (divide by highest power of x or conjugate) for complex expressions.

Worked Examples

• lim(x→∞) [1/x²] = 0 for bottom-heavy functions.
• lim(x→∞) [8/(3x+4)] = 0; denominator dominates.
• lim(x→∞) [(8x–5)/(2x+3)] = 4; equal degrees, divide coefficients.
• lim(x→∞) [(5–7x³)/(3x+5x³+9)] = –7/5; highest degrees, divide leading coefficients.
• lim(x→∞) [√(16x²–8)/(2x–5)] = 2; factor out and simplify under the root.
• lim(x→∞) [√(9x⁶–x²)/(3x³+1)] = 1; highest powers, roots simplify numerator and denominator.
• lim(x→∞) [√(9x²+x)–3x] = 1/6; use conjugate and simplify.

Special Functions

• lim(x→∞) [arctan(x)] = π/2 (90 degrees); arctangent approaches its horizontal asymptote.
• lim(x→–∞) [arctan(x)] = –π/2 (–90 degrees).
• lim(x→∞) [e^(–x)] = 0; exponential decay.
• lim(x→–∞) [e^(x)] = 0; exponential approaches 0 as x goes negative.

Key Terms & Definitions

• Limit at Infinity — The value a function approaches as x grows without bound.
• Horizontal Asymptote — Line y = L that a function approaches as x→±∞.
• Vertical Asymptote — Line x = a where function grows without bound.
• Bottom-Heavy Function — Degree of denominator > numerator; limit is 0 at infinity.
• Arctangent (arctan) — Inverse tangent function; approaches π/2 or –π/2 at infinity.
• Conjugate — An expression used to simplify radicals in limits.

Action Items / Next Steps

• Practice more limit problems, especially rational and radical cases.
• Review properties of asymptotes and exponential/logarithmic limits.
• Memorize key behaviors (arctan, exponential) as x approaches infinity.