Overview

The concept of limit of a function is one of the fundamental ideas that distinguishes calculus from algebra and trigonometry.

Definition

Informal Definition of Limit

Let ( f(x) ) be defined on an open interval about ( $ x_0 $ ), except possibly at ( $ x_0 $ ) itself. If ( f(x) ) gets arbitrarily close to ( L ) for all ( x ) sufficiently close to ( $ x_0 $ ), we say that ( f ) approaches the limit ( L ) as ( x ) approaches ( $ x_0 $ ), and we write

\[\lim_{x \to x_0} f(x) = L.\]

Theorem 1

Properties of Limits

The following rules hold if ( \(\lim_{x \to c} f(x) = L\) ) and ( \(\lim_{x \to c} g(x) = M\) ) ( L and M real numbers).

  1. Sum Rule: \(\lim_{x \to c} [f(x) + g(x)] = L + M\)

  2. Difference Rule: \(\lim_{x \to c} [f(x) - g(x)] = L - M\)

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

  4. Constant Multiple Rule: \(\lim_{x \to c} k f(x) = kL \quad \text{(any number } k)\)

  5. Quotient Rule: \(\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}, \quad M \neq 0\)

  6. Power Rule: If ( m ) and ( n ) are integers, then \(\lim_{x \to c} [f(x)]^{m/n} = L^{m/n},\) provided (\(L^{m/n}\)) is a real number.