End Behavior
What happens to a function when \(x\) becomes trillions and trillions? This is the limit at infinity: \(\lim_{x \to \infty} f(x)\). If the function settles down to a specific height, that height is a Horizontal Asymptote.
The Degree Rule
For rational functions (fractions with polynomials):
- Bottom Heavy: Limit is 0.
- Top Heavy: Limit is infinity (no asymptote).
- Equal Degree: Limit is the ratio of the leading coefficients.
Worked Examples
Example 1: Balancing Powers
Find \(\lim_{x \to \infty} \frac{3x^2 + 5}{x^2 - 10}\).
- The highest power on top and bottom is \(x^2\).
- The coefficients are 3 and 1.
- Result: 3
Example 2: Vanishing Functions
Find \(\lim_{x \to \infty} \frac{10}{x}\).
- As \(x\) gets bigger, the fraction gets smaller.
- Result: 0
The Bridge to Quantum Mechanics
In Quantum Mechanics, we require that every physical wavefunction must go to Zero at Infinity: \(\lim_{x \to \pm\infty} \psi(x) = 0\). If a wavefunction didn't go to zero, it would mean there is a chance the particle is "at infinity," which would make it impossible to normalize the probability to 1. This simple limit rule is the reason why particles stay "bound" to atoms instead of flying off into the void. A particle that is "trapped" is a particle whose wavefunction satisfies this limit at infinity.