Chapter 1 Limit and their Properties

Section 1.2 Finding Limits Graphically and Numerically I. Different Approaches A. Numerical Approach 1. Construct a table of values for x and f(x). 2. Use small intervals for x to estimate the limit that f(x) approaches. B. Graphical Approach 1. Use the table to plot points. 2. Draw a graph by hand or use a calculator. C. Analytical Approach 1. Use algebra or calculus 2. Plug in the value for x that it approaches

II. Common types of behavior associated with the nonexistence of a limit. A. f(x) approaches a different number from the right side of c than it approaches from the left side. B. f(x) increases or decreases without bound as x approaches c. C. f(x) oscillates between 2 fixed values as x approaches c. III. Definition of a limit. A. Let f be a function defined at an open interval containing c (except possibly at c) and let L be a real number. The statement f(x) = L Means that for each ε > 0 there exists a δ > 0 such that if 0 < |x-c| < δ, then |f(x) – L| < ε

Section 1.3: Evaluating Limits Analytically A. Direct Substitution 1. Factor and Cancel 2. Rationalize Numerator 3. B. Properties of Limits 1. Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits and a. Scalar multiple: b. Sum or difference: c. Product: d. Quotient:provided K ≠ 0 e. Power: n

C. Two Special Trigonometric Limits 1. 2.

Section 1.4 Continuity and One-Sided Limits A. Definition of Continuity 1. Continuity at a point : A function f is continuous at c if the following 3 conditions are met a.f(c) is defined b. exists c. 2. Continuity on an open interval: A function is continuous on an open interval (a,b) if it is continuous at each point in the interval 3. Removable and non-removable discontinuities.

B. One-Sided Limits 1. Existence of a Limit: Left f be a function and let c and L be real number. The limit f(x) as x approaches c is L if and only if and C. Continuity on a Closed Interval 1. Definition: A function f is continuous on the closed interval if it is continuous on the open interval (a,b) and and D. Intermediate Value Theorem 1. Definition: If f is continuous on the closed interval and K is any number between f(a) and f(b), then there is at least one number c in such that f(c)=K.

Section 1.5 – Infinite Limits I. Infinite Limits A. Let f be a function that is defined at every real number in some open interval containing c (except possibly c itself). The statement means that for each M > 0 there exists a  > 0 such that whenever Similarly, the statement means that for each N 0 such that f(x) < N whenever.To define the infinite limit from the left, replace by c –  < x < c. To define the infinite limit from the right, replace by c < x < c + . B. lim f(x) = does not mean that the limit exists, it tells you how the limit fails to exist by demonstrating the unbounded behavior of f(x) as x approaches c.

II. Vertical Asymptotes A. If f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x = c is a vertical asymptote of the graph of f. B. Let f and g be continuous on an open interval containing c. If f(c) 0, g(c) = 0, and there exists an open interval containing c such that g(x) 0 for all x c in the interval, then the graph of the function given by has a vertical asymptote at x = c. (Vertical asymptote occurs at a number where the denominator is 0)

III. Properties of Infinite Limits Let c and L be real numbers and let f and g be functions such that and 1.Sum or difference: 2.Product:L > 0 L < 0 3.Quotient: Similar properties hold for one-sided limits and for functions for which the limit of f(x) as x approaches c is