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1.15 Connecting Limits at Infinity and Horizontal Asymptotes

Keywords

Limits at Infinity 无穷远极限 • Horizontal Asymptote 水平渐近线

End Behavior 终端行为 • Relative Magnitude 相对量级 • Growth Rates 增长速率

Limits at Infinity

While infinite limits (Section 1.14) describe \(y\) going to infinity as \(x\) approaches a number, limits at infinity describe what happens to \(y\) as \(x\) itself goes to \(\infty\) or \(-\infty\).

核心要点：Horizontal Asymptotes (LIM-2.D.3 & LIM-2.D.4)

The line \(y = L\) is a horizontal asymptote of the graph of \(f\) if:

\[ \lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L \]

Limits at infinity describe the end behavior of a function—where the graph "settles down" as you move far to the left or right.

Limits of Rational Functions

For rational functions \(f(x) = \frac{P(x)}{Q(x)}\), the limit at infinity depends on the degree (highest power) of the numerator and denominator:

  1. Bottom-Heavy: If the degree of the denominator is greater, the limit is \(0\). Example: \(\lim\limits_{x \to \infty} \frac{x+1}{x^2-3} = 0 \implies\) Horizontal Asymptote at \(y=0\).

  2. Equal Degrees: If the degrees are the same, the limit is the ratio of the leading coefficients. Example: \(\lim\limits_{x \to \infty} \frac{5x^2+1}{2x^2-3} = \frac{5}{2} \implies\) Horizontal Asymptote at \(y=2.5\).

  3. Top-Heavy: If the degree of the numerator is greater, the limit is \(\infty\) or \(-\infty\) (no horizontal asymptote).

Example: \(f(x) = \frac{2x^2}{x^2+1}\)

As \(x \to \infty\), the \(x^2\) terms dominate, and the function approaches \(2/1 = 2\).

Relative Magnitudes and Growth Rates

According to LIM-2.D.5, we can determine limits at infinity by comparing how fast functions grow. Some functions grow much faster than others as \(x \to \infty\).

核心要点：Hierarchy of Growth Rates

As \(x \to \infty\), the following list ranks common functions from slowest growing to fastest growing:

\[ \ln(x) \ll x^n \ll e^x \ll x! \ll x^x \]
  • Logs are the slowest.
  • Polynomials are in the middle (higher degrees grow faster).
  • Exponentials grow faster than any polynomial.

Example: Evaluate \(\lim\limits_{x \to \infty} \frac{x^{100}}{e^x}\). Even though \(x^{100}\) seems huge, \(e^x\) grows exponentially faster. Therefore, the denominator will eventually dwarf the numerator, and the limit is \(0\).

Extension: Can a graph cross its asymptote?

A common misconception is that a graph can never cross an asymptote. While this is usually true for vertical asymptotes, a graph can cross a horizontal asymptote many times. A horizontal asymptote only describes what happens as \(x\) becomes extremely large; it doesn't restrict the function's behavior near the origin.

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