Limits and Continuity MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Fall 2018
Objectives After this lesson we will be able to: Determine the left-hand and right-hand limits of functions. Determine if a limit of a function exists as a approaches a real number a. Use the properties of limits to determine limits of functions. Find the limits of rational functions. Identify continuous functions.
Introduction Remark: Limits underlie most of the topics of calculus. It is important that you understand what limits mean, not just how to find them. Today we will introduce numerical, graphical, and algebraic approaches to finding the limit.
Notation Suppose f (x) is a function defined on an interval containing a (but not necessarily at a). lim f (x) is read as the limit of f (x) as x approaches a from x a the left. lim f (x) is read as the limit of f (x) as x approaches a from x a + the right. lim f (x) is read as the limit of f (x) as x approaches a. x a
Notation Suppose f (x) is a function defined on an interval containing a (but not necessarily at a). lim f (x) is read as the limit of f (x) as x approaches a from x a the left. lim f (x) is read as the limit of f (x) as x approaches a from x a + the right. lim f (x) is read as the limit of f (x) as x approaches a. x a Remark: the first two are called one-sided limits.
Example: Graphical Approach lim x 1 x 2 + x x 2 x 2 0.5 0.4 0.3 0.2 0.1 2.0 1.5 1.0 0.5
Example: Numerical Approach lim x 1 x 2 + x x 2 x 2 x f (x) 1.1 0.354839 1.01 0.335548 1.001 0.333555 1.0001 0.333356
Example: Graphical Approach lim x 1 + x 2 + x x 2 x 2 0.5 0.4 0.3 0.2 0.1 2.0 1.5 1.0 0.5
Example: Numerical Approach lim x 1 + x 2 + x x 2 x 2 x f (x) 0.9 0.310345 0.99 0.331104 0.999 0.333111 0.9999 0.333311
Informal Definition Definition (Left-Hand Limits) If the values of function y = f (x) get closer and closer to some number L as values of x, that are smaller than some number a, get closer and closer to a, then we say that L is the limit of f (x) as x approaches a from the left. We write lim f (x) = L. x a Definition (Right-Hand Limits) If the values of function y = f (x) get closer and closer to some number M as values of x, that are larger than some number a, get closer and closer to a, then we say that M is the limit of f (x) as x approaches a from the right. We write lim f (x) = M. x a +
Example Find the following limits, if they exist. 6 5 4 3 2 1 1 2 3 4 5 6 lim x 2 = lim x 2 + = lim x 4 = lim x 4 + = lim x 0 = lim x 0 + =
Example Find the following limits, if they exist. 6 5 4 3 2 1 1 2 3 4 5 6 lim x 2 = 4 lim x 2 + = 2 lim x 4 = 2 lim x 4 + = 4 lim f (x) x 0 = DNE lim x 0 + = 0
Polynomials and Limits Definition A polynomial function is a function of the form P(x) = a n x n + a n 1 x n 1 + + a 2 x 2 + a 1 x + a 0 where each exponent is a positive integer and a n, a n 1,..., a 0 are real numbers. If P is a polynomial and c is a real number, then lim P(x) x c = P(c) lim P(x) x c + = P(c) lim x c P(x) = P(c).
Example: Unbounded Limits lim x 2 x 2 + x x 2 x 2 20 10 1.5 2.0 2.5 3.0 10 20
Example: Unbounded Limits lim x 2 x 2 + x x 2 x 2 20 10 1.5 2.0 2.5 3.0 10 20 Conclusion: lim x 2 x 2 + x x 2 x 2 =
Example: Numerical Approach lim x 2 x 2 + x x 2 x 2 x f (x) 1.9 19. 1.99 199. 1.999 1999. 1.9999 19999.
Example: Numerical Approach Conclusion: lim x 2 lim x 2 x 2 + x x 2 x 2 x f (x) 1.9 19. 1.99 199. 1.999 1999. 1.9999 19999. x 2 + x x 2 x 2 =.
Unbounded Limits Definition (+ Limits) If f (x) increases without bound as x approaches a from the left (or from the right), then we say that f (x) approaches positive infinity. We write lim x a x a f (x) = + or lim f (x) = +. + Definition ( Limits) If f (x) decreases without bound as x approaches a from the left (or from the right), then we say that f (x) approaches negative infinity. We write lim x a x a f (x) = or lim f (x) =. +
Limits: Considering Both Sides Definition If the values of f (x) get closer and closer to some number L as values of x that are smaller than some number c and values of x that are larger than c get closer and closer to c (but not equal to c), then L is the limit of f (x) as x approaches c. For a function f and a real number c, if lim f (x) = L and lim f (x) = L x c x c + then we write lim f (x) = L. x c
Existence of a Limit 1. We say that the limit of a function f approaches a real number c exists if and only if lim x c f (x) = L, where L is a real number. 2. If lim f (x) lim f (x), then lim f (x) does not exist. x c x c + x c 3. If lim f (x) = + or lim f (x) =, then lim f (x) does x c x c x c not exist.
Simple Properties Theorem For any constant c and any real number a, lim c = c. x a
Simple Properties Theorem For any constant c and any real number a, lim c = c. x a Theorem For any real number number a, lim x = a. x a
Simple Properties Theorem For any constant c and any real number a, lim c = c. x a Theorem For any real number number a, lim x = a. x a Theorem For any real number number a, lim x n = a n. x a
The Arithmetic of Limits Theorem Suppose that lim f (x) = L and lim g(x) = M and that c is a x a x a constant. Then the following hold: 1. lim [c f (x)] = c lim f (x) = cl. x a x a 2. lim [f (x) ± g(x)] = lim f (x) ± lim g(x) = L ± M. x a x a x a [ ] [ ] 3. lim [f (x) g(x)] = lim f (x) lim g(x) = LM. x a x a x a f (x) 4. lim x a g(x) = lim x a f (x) lim x a g(x) = L, provided M 0. M 5. lim x a n f (x) = n L, if n L is defined.
Summary of the Rules of Limits For functions composed of additions, subtractions, multiplications, divisions, powers, and roots, limits may be evaluated by direct substitution, provided that the resulting expression is defined. lim f (x) = f (c) x c The same thing can be said for one-sided limits.
Indeterminate Forms Definition A limit expression of the type f (x) lim x a g(x) is called an indeterminate form of type 0/0 if lim x a f (x) = 0 and lim x a g(x) = 0.
Example: Indeterminate Forms lim x 1 x 2 + x x 2 x 2 = lim x 1 = lim x 1 x(x + 1) (x 2)(x + 1) x x 2 = 1 3
Limits at Infinity Use numerical and/or graphical evidence to conjecture the values of the following limits. 1 1. lim x x 1 2. lim x x 5 1 x 0 5 10 10 5 0 5
Horizontal Asymptotes Definition The function f (x) has a horizontal asymptote at y = L if lim f (x) = L or lim f (x) = L. x x +
Finding Horizontal Asymptotes Let f be the rational function f (x) = a nx n + a n 1 x n 1 + + a 1 x + a 0 b m x m + b m 1 x m 1 + + b 1 x + b 0, where a n 0 and b m 0. The graph of f has zero or one horizontal asymptote determined by comparing n and m. 1. If n < m, the graph of f has the line y = 0 as a horizontal asymptote. 2. If n = m, the graph of f has the line y = a n b m as a horizontal asymptote. 3. If n > m, the graph of f has no horizontal asymptote.
Example Find the horizontal and vertical asymptotes (if any) for the following function. f (x) = 6x 2 11x + 3 6x 2 7x 3
Example Find the horizontal and vertical asymptotes (if any) for the following function. f (x) = f (x) = 6x 2 11x + 3 6x 2 7x 3 (3x 1)(2x 3) (3x + 1)(2x 3) = 3x 1 3x + 1 if x 3/2. The domain is the set of real numbers {x x 3/2, x 1/3}. There is a vertical asymptote at x = 1 3. There is a horizontal asymptote at y = 3 3 = 1.
Intuitive Idea of Continuity A process or an item can be described as continuous if it exists without interruption. The mathematical definition of continuity is more technical than intuitive, but is based on the concept of the limit.
Graphical Examples of Discontinuities (1 of 4) y a f (a) is undefined. The graph has a hole at x = a. This is called a removeable discontinuity.
Graphical Examples of Discontinuities (2 of 4) y a f (a) is defined but the lim x a f (x) does not exist. This is sometimes called a jump discontinuity.
Graphical Examples of Discontinuities (3 of 4) y f a a f (a) is defined and the lim f (x) exists, but lim f (x) f (a). This x a x a is also called a removeable discontinuity.
Graphical Examples of Discontinuities (4 of 4) y a The lim x a f (x) does not exist (is not finite).
Definition of Continuity Definition A function f is continuous at x = a when 1. f (a) is defined, 2. lim x a f (x) exists, and 3. lim x a f (x) = f (a). Otherwise f is said to be discontinuous at x = a.
Definition of Continuity Definition A function f is continuous at x = a when 1. f (a) is defined, 2. lim x a f (x) exists, and 3. lim x a f (x) = f (a). Otherwise f is said to be discontinuous at x = a. Theorem All polynomial functions are continuous at all real numbers. Every rational function is continuous except where the denominator is zero.
Example Suppose the formula for the amount owed for state income tax is { 0.12x if 0 < x < 20, 000, T (x) = a + 0.16(x 20, 000) if x 20, 000. where x is the income of the taxpayer. Determine value for the constant a that makes the function T continuous everywhere.
Arithmetic of Continuous Functions Continuous Functions If functions f and g are continuous at x = c, then the following are also continuous at x = c: f ± g a f (for any constant a) f g f g (if g(c) 0) f (g(x)) (if f is continuous at g(c))
Continuity on Intervals A function is continuous on an open interval (a, b) if it it continuous at each point in the interval. A function is continuous on a closed interval [a, b] if it it continuous at each point in the open interval (a, b) and if lim f (x) x a + = f (a) lim f (x) x b = f (b). A function that is continuous on the real number line (, ) is said to be continuous everywhere.