Study Notes on Limits, Infinity, and Intermediate Value Theorem

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Today's Goals

1. Discuss Limits and Infinity
   • Understanding the concept of limits and infinity.
   • Evaluating limits at infinity for rational functions.
2. Define Rational Functions
   • Defined as the ratio of two polynomials: $f(x) = \frac{F(x)}{G(x)}$ where (F) and (G) are polynomials.
3. Introduce Intermediate Value Theorem
   • Apply this theorem to find zeros of functions.

Limits and Infinity

Visualizing Limits

• Example: A function approaching a horizontal line, where:
  • As (x \to \infty), (f(x) \to 3)
  • As (x \to -\infty), (f(x) \to -3)
• Notation for Limits:
  • Limit of (f(x)) as (x \to \infty) is equal to 3:
    $\lim_{x \to \infty} f(x) = 3$
  • Limit of (f(x)) as (x \to -\infty) is equal to -3:
    $\lim_{x \to -\infty} f(x) = -3$
• Infinity is a tendency, not a fixed point.

Definition of Limit at Infinity

• Definition:
  • Suppose there exists a real number (l) such that for all (\epsilon > 0), there exists an (N) such that if (x > N), then (|f(x) - l| < \epsilon).
  • Limits express that as (x) becomes large enough, the function approaches (l).

Understanding Infinity in Limits

• Notation for positive & negative infinity:
  • Positive Infinity: can use (\infty) or (+\infty)
  • Negative Infinity: always use (-\infty)

Practical Examples

Exponential Functions

• Example 1: If (a > 1), then:
  • $\lim_{x \to \infty} a^x = +\infty$
  • $\lim_{x \to -\infty} a^x = 0$
• As (x \to -\infty), function becomes negligible (goes to zero).
  • Reasoning: (a^{-x} = \frac{1}{a^x}), and since (a^x) grows, (\frac{1}{a^x}) diminishes to zero.

Functions with Base Between 0 and 1

• Example 2: If (0 < a < 1), then:
  • $\lim_{x \to \infty} a^x = 0$
  • For negative input, it behaves as in previous example, approaching (+\infty) when negative powers yield greater than 1.

Polynomial Functions

• If (n > 0), then:
  • $\lim_{x \to \infty} x^n = +\infty$
• If (n < 0):
  • $\lim_{x \to \infty} x^{-n} = 0$

Behavior Depending on Even or Odd Exponents

• Example: If (n) is even versus odd for (x \to -\infty):
  • Even (n): $\lim_{x \to -\infty} x^n = +\infty$
  • Odd (n): $\lim_{x \to -\infty} x^n = -\infty$

Evaluating Limits of Rational Functions

Theorem for Rational Functions

• The limit as (x \to \pm \infty) for rational functions is determined by the highest degree terms:
  • Form: ( \frac{an x^n + …}{bn x^m + …} )
  • Focus only on leading terms:
    $\lim<em>{x \to \infty} \frac{a</em>n x^n}{b<em>m x^m} = \frac{a</em>n}{b_m} x^{n-m}$

Simplifying Rational Functions

• Example: For limit $\lim_{x \to \infty} \frac{4x^6 + 5x + 1}{2x^3 - x^2 + 1}$:
  • Apply the theorem:
  • Focus on leading terms: $\frac{4x^6}{2x^3} = 2x^{3}$
  • Thus: $\lim_{x \to \infty} 2x^3 = +\infty$
• Example when leading terms are the same:
  • For $\frac{x^2}{x^2}$, result equals 1.

Intermediate Value Theorem

Definition

• States that for any continuous function (f) over the interval ([a,b]):
  • If (f(a) < m < f(b)), then there exists a (c \in (a,b)) such that (f(c) = m).
• Visually: Draw horizontal lines between (f(a)) and (f(b)); they intersect the curve.

Application

• Used to find zeros of functions:
  • If (g(a) > 0) and (g(b) < 0), then there is a root (zero) between (a) and (b).