Chapter 2:Limits, Calc 1

Definition 2.5 (Formal Limit Definition)

Suppose f is defined on (c, a) U (a, b) where c < a and b > a. Then lim(x->a) f(x) = L if for every epsilon > 0 there exists a delta > 0 such that |f(x) - L| < epsilon whenever |x - a| < delta.

Theorem 2.9 (Uniqueness of Limits)

If lim(x->a) f(x) = L and lim(x->a) f(x) = M, then L = M.

Theorem 2.12 (Arithmetic of Limits - Sum)

If lim(x->a) f(x) = L and lim(x->a) g(x) = M, then lim(x->a) (f(x) + g(x)) = L + M.

Theorem 2.12 (Arithmetic of Limits - Difference)

If lim(x->a) f(x) = L and lim(x->a) g(x) = M, then lim(x->a) (f(x) - g(x)) = L - M.

Theorem 2.12 (Arithmetic of Limits - Product)

If lim(x->a) f(x) = L and lim(x->a) g(x) = M, then lim(x->a) f(x)g(x) = LM.

Theorem 2.12 (Arithmetic of Limits - Quotient)

If lim(x->a) f(x) = L and lim(x->a) g(x) = M, then lim(x->a) (f(x) / g(x)) = L / M, provided M != 0.

Theorem 2.12 (Arithmetic of Limits - Constant Multiple)

If lim(x->a) f(x) = L, then lim(x->a) cf(x) = cL for any constant c.

Theorem 2.12 (Arithmetic of Limits - Inequality)

If lim(x->a) f(x) = L and lim(x->a) g(x) = M, and f(x) <= g(x) near a, then L <= M.

Definition 2.13 (Polynomial and Rational Functions)

A polynomial is p(x) = a_n*x^n + … + a_0. A rational function is r(x) = p(x) / q(x) where p and q are polynomials.

Corollary 2.14 (Limits of Rational Functions)

Let p(x) and q(x) be polynomials and a be a real number such that q(a) != 0. Then lim(x->a) (p(x) / q(x)) = p(a) / q(a).

Theorem 2.17 (The Squeeze Theorem)

Suppose f(x)

Trigonometric Limit Identity 1

lim(x->0) (sin(x) / x) = 1

Definition 2.20 (Right-Hand Limit)

Suppose f is defined on (a, b). Then lim(x->a+) f(x) = L if for every epsilon > 0 there exists delta > 0 such that |f(x) - L| < epsilon whenever 0 < x - a < delta.

Definition 2.21 (Left-Hand Limit)

Suppose f is defined on (c, a). Then lim(x->a-) f(x) = L if for every epsilon > 0 there exists delta > 0 such that |f(x) - L| < epsilon whenever 0 < a - x < delta.

Definition (Limit at Infinity)

If f is defined on (a, infinity), lim(x->infinity) f(x) = L if f(x) approaches L as x approaches infinity.

Fundamental Limit at Infinity

lim(x->infinity) (1 / x) = 0

Distance Inequality: Triangle Inequality

|a - b| <= |a - c| + |c - b|

Distance Inequality: Sum

|a + b| <= |a| + |b|