The Role Played by Infinity in Limits

Types of Infinity in Limits

Limits Approaching Infinity:

• This occurs when the variable within a function approaches infinity. The notation is \( \lim_{x \to \infty} f(x) \) or \( \lim_{x \to -\infty} f(x) \).
• The limit evaluates how the function behaves as the variable grows larger and larger (positively or negatively).

Limits Equaling Infinity:

• This happens when the function itself grows without bound as the variable approaches a certain finite value. The notation is \( \lim_{x \to a} f(x) = \infty \) or \( \lim_{x \to a} f(x) = -\infty \).
• It often indicates a vertical asymptote at \( x = a \).

Evaluating Limits Involving Infinity

Polynomial Functions:

• For high-degree polynomials, as \( x \) approaches infinity, the behavior of the function is dominated by the term with the highest power.
• Example: \( \lim_{x \to \infty} (3x^4 – 2x^3 + 5) = \infty \).

Rational Functions:

• The behavior is determined by the degrees of the numerator and denominator.
• If the degree of the numerator is greater, the limit is infinity; if less, the limit is zero; if equal, the limit is the ratio of the leading coefficients.
• Example: \( \lim_{x \to \infty} \frac{2x^2 + 3x}{5x^2 + 7} = \frac{2}{5} \).

Exponential Functions:

• Exponential functions grow faster than polynomial functions.
• Example: \( \lim_{x \to \infty} e^x = \infty \).

Trigonometric Functions:

• Trigonometric functions do not have limits as \( x \) approaches infinity since they oscillate.
• Example: \( \lim_{x \to \infty} \sin(x) \) does not exist.

Logarithmic Functions:

• They grow more slowly than polynomial functions. As \( x \) approaches infinity, logarithmic functions approach infinity but at a slower rate.
• Example: \( \lim_{x \to \infty} \ln(x) = \infty \).

Special Considerations

• Indeterminate Forms: Forms like \( \frac{\infty}{\infty} \) or \( 0 \cdot \infty \) are indeterminate, requiring additional techniques like L’Hôpital’s Rule or algebraic manipulation for evaluation.
• Behavior Near Vertical Asymptotes: When a function approaches a vertical asymptote, the limit typically approaches infinity or negative infinity.