Limit Concepts and Theorems

Finding Limits

• The limit of the product of two functions can be found using the property:
  $\lim<em>{x \to a} [f(x) \cdot g(x)] = \lim</em>{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
• Example:
• $\lim<em>{x \to \infty} [2 + \lim</em>{x \to \infty} \frac{1}{x}]$
• As $x$ approaches infinity, $\frac{1}{x}$ approaches 0. Thus, the entire limit equals:
  $2 + 0 = 2$

Horizontal Asymptote

• A horizontal asymptote exists if:
  $\lim_{x \to \infty} f(x) = l$
• If $l$ is a constant, $y = l$ is a horizontal asymptote.
• Example: For the function $2 + \frac{1}{x}$, as $x$ approaches infinity, the horizontal asymptote is at $y = 2$.

Behavior of Functions at Infinity

• To study the function's behavior at infinity, consider both ends:
• Functions may approach a finite limit or infinity.
• Example with $\frac{1}{x}$:
• $\lim<em>{x \to \infty} \frac{1}{x} = 0$, and $\lim</em>{x \to -\infty} \frac{1}{x} = 0$.

Squeeze Theorem

• Useful for finding limits when a function is bounded by two others:
• If $f(x) \leq h(x) \leq g(x)$ and $\lim<em>{x \to a} f(x) = \lim</em>{x \to a} g(x) = l$ then $\lim_{x \to a} h(x) = l$.
• Example using sine:
• $-1 \leq \sin(x) \leq 1$ implies:
  $\frac{-1}{x} \leq \frac{\sin(x)}{x} \leq \frac{1}{x}$; thus, $\lim_{x \to \infty} \frac{\sin(x)}{x} = 0$.

General Rules for Limits at Infinity

• For polynomial functions:
• $\lim_{x \to \infty} x^n = \infty$ (for n > 0).
• $\lim_{x \to \infty} x^n = 0$ (for n < 0).
• Products and constant multiples:
• $\lim<em>{x \to \infty} c \cdot x^n = c \cdot \lim</em>{x \to \infty} x^n$.

Finding Rational Function Limits

• When dealing with rational functions:
• The limit is found by dividing leading coefficients if degrees are equal:
  • For $\frac{a<em>n x^n + ext{lower terms}}{b</em>m x^m + ext{lower terms}}$,
  • if $n=m$, then $\frac{a<em>n}{b</em>m}$.
• Example: For $\frac{3x^3 + 2x - 2}{4x^3 + 2}$:
• $\lim_{x \to \infty} = \frac{3}{4}$.

Summary of Rules

• Rule of Polynomials: $\lim_{x \to a} c = c$
• Addition and Subtraction of Limits: $\lim{x \to a} [f(x) + g(x)] = \lim{x \to a} f(x) + \lim_{x \to a} g(x)$
• Product of Limits: $\lim{x \to a} [f(x) \cdot g(x)] = \lim{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
• Quotient of Limits: If $g(x) \neq 0$, \$\lim{x \to a} \frac{f(x)}{g(x)} = \frac{\lim{x \to a} f(x)}{\lim_{x \to a} g(x)}$.