Skip to content

1.5 Determining Limits Using Algebraic Properties

Keywords

Limit Laws 极限法则 • Direct Substitution 直接代入法

Sum and Difference Rules 和差法则 • Product and Quotient Rules 积商法则

Composite Functions 复合函数 • Analytic Evaluation 解析法求极限

Basic Limit Properties

If the limits \(\lim\limits_{x \to c} f(x) = L\) and \(\lim\limits_{x \to c} g(x) = M\) exist (where \(L\) and \(M\) are real numbers), the following properties hold:

Core Concept: Algebraic Properties of Limits

Let \(k\) be a constant. The following rules allow us to break down complex limits into simpler parts:

  1. Sum/Difference Rule: \(\lim\limits_{x \to c} [f(x) \pm g(x)] = \lim\limits_{x \to c} f(x) \pm \lim\limits_{x \to c} g(x) = L \pm M\)

  2. Constant Multiple Rule: \(\lim\limits_{x \to c} [k \cdot f(x)] = k \cdot \lim\limits_{x \to c} f(x) = k \cdot L\)

  3. Product Rule: \(\lim\limits_{x \to c} [f(x) \cdot g(x)] = [\lim\limits_{x \to c} f(x)] \cdot [\lim\limits_{x \to c} g(x)] = L \cdot M\)

  4. Quotient Rule: \(\lim\limits_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim\limits_{x \to c} f(x)}{\lim\limits_{x \to c} g(x)} = \frac{L}{M}\), provided \(M \neq 0\).

  5. Power/Root Rule: \(\lim\limits_{x \to c} [f(x)]^n = L^n\) (if \(n\) is a fraction, \(L\) must be positive for even roots).

Direct Substitution

For many well-behaved functions, such as polynomials and rational functions (where the denominator is not zero), the limit as \(x \to c\) is simply the value of the function at \(c\).

\[ \lim_{x \to c} f(x) = f(c) \]

This is because these functions are "continuous" (a concept we will explore further in Section 1.6).

Example: Evaluating a Polynomial

Evaluate \(\lim\limits_{x \to 2} (x^2 + 3x - 1)\). Using the properties:

\[ \lim_{x \to 2} x^2 + \lim_{x \to 2} 3x - \lim_{x \to 2} 1 = (2)^2 + 3(2) - 1 = 4 + 6 - 1 = 9 \]

Limits of Composite Functions

For a composite function \(f(g(x))\), we evaluate the limit by "working our way in."

Core Concept: Limits of Composite Functions

If \(f\) is continuous at the limit of \(g(x)\), then:

\[ \lim_{x \to c} f(g(x)) = f\left( \lim_{x \to c} g(x) \right) \]

  1. Find the limit of the inner function: \(L = \lim\limits_{x \to c} g(x)\).

  2. Evaluate the outer function at that limit: \(f(L)\).

Quiz