C hapter 3 Limits and Their Properties

Section 3.1 A Preview of Calculus

What is Calculus  Calculus is the mathematics of change--- velocities and accelerations  Calculus is also the mathematics of tangent lines, slopes, areas, volumes, arc lengths, centroids, curvatures, and a variety of other concepts that have enabled scientists, engineers, and economists to model real-life situations

The Difference Between Precalculus and Calculus  Precalculus deals with velocities, acceleration, tangent lines, slopes, and so on, there is a fundamental difference between precalculus.  Precalculus is more statics, whereas calculus is more dynamic

Precalculus vs. Calculus Precalculus Calculus  An object traveling at a constant velocity can be analyzed with precalculus  The slope of a line can be analyzed with precalculus  A tangent line to a circle can be analyzed with precalculus  The area of a rectangle can be analyzed with precalculus  To analyze the velocity of an accelerating object, you need calculus  To analyze the slope of a curve you need calculus  To analyze a tangent line to a general graph, you need calculus  To analyze the area under a general curve, you need calculus

The Tangent Line Problem Given a function f and a point P on its graph, you are asked to find an equation of the tangent line to the graph at point P Excluding a vertical tangent line, the problem is equivalent to finding the slope of the tangent line at P You can approximate this slope by using a line through the point of tangency and a second point on the curve. Such a line is called a secant line

The Tangent Line Problem Cont’d If P(c, f(c)) is the point of tangency and Q(c + ∆x, f(c + ∆x) is a second point on the graph of f The slope of the secant line through these two points is m sec = (f(c + ∆x) - f(c)) / (c + ∆x – c) or m sec = (f(c + ∆x) - f(c)) / (∆x)

The Tangent Line Problem Cont’d As point Q approaches point P, the slope of the secant line approaches the slope of the tangent line. When such a “limiting position” exists, the slope of the tangent line is said to be the limit of the slope of the secant line. EXAMPLE The following points lie on the graph of f(x) = x 2 Q 1 (1.5, f(1.5), Q 2 (1.1,f(1.1), Q 3,(1.01,f(1.01),Q 4 (1.001, f(1.001) and Q 5 (1.0001, f(1.0001) Estimate the slope of the tangent line of f at point P

The Area Problem Find the area of a plane region bounded by the graphs of functions. This problem can be solved with a limit process. In this case, the limit process is applied to the area of a rectangle to find the area of a general region. Consider the region bounded by the graph of the function y = f(x), the x-axis, and the vertical lines x = a and x= b You can approximate the area of the region with several rectangular regions, the more rectangles, the better the approximation Your goal is to determine the limit of the sum of the areas of the rectangles as the number of rectangles increase without bound.

The Area Problem - Example Consider the region bounded by the graphs of f(x) = x 2, y = 0, and x = 1 First inscribe a set of rectangles, then circumscribe a set of rectangles. Find the sum of the areas of each set of rectangles Use your results to approximate the area of the region

Section 3.2 Finding Limits Graphically and Numerically

An Introduction to Limits Let f(x) = (x 3 – 1)/(x – 1), however x ≠ 1 Since you don’t know the behavior of the graph at x = 1, use two sets of values, one set that approaches 1 from the left and one set that approaches 1 from the right. Make a table. The graph of f is a parabola with a gap Although x cannot equal 1, you can move x arbitrarily close to 1, and as a result f(x) moves arbitrarily close to ????

An Introduction to Limits – Cont’d Using limit notation you can write: lim f(x) = 3 x → 1 Therefore, if f(x) becomes arbitrarily close to a single L as x approaches c from either side, the limit of f(x), as x approaches c, is L and is written lim f(x) = L x → c

Example Estimate the limit of each function 1. f(x) = x/((x + 1) ½ - 1) at 0 2. f(x) = 1/x 2 at 0 3. f(x) = x 2 at 2

Section 3.3 Evaluating Limits Analytically

Some Basic Limits Let b and c be real numbers and let n be a positive integer lim b = b x → c lim x = c x → c lim x n = c n x → c

Properties of Limits Let b and c be real numbers and let n be a positive integer, and let f and g be function with the following limits lim f(x) = L and lim g(x) = K x → c x → c 1. Scalar multiple: lim [bf(x) = bL x → c 2. Sum or difference: lim [f(x) ± g(x)] = L ± K x → c 3. Product: lim [f(x) g(x)] = LK x → c 4. Quotient: lim [f(x)/g(x)] = L/K provided K ≠ 0 x → c 5. Power: lim [f(x) n )] = L n x → c

Example Find the limit of each polynomial 1. lim (4x 2 + 3) x → 2 2. lim (x 2 + x + 2) x → 1

Limits of Polynomial and Rational Functions If p is a polynomial function and c is a real number, then lim p(x) = p(c) x → c If r is a rational function given by r(x) = p(x)/q(x) and c is a real number such that q(c) ≠ 0, then lim r(x) = r(c) = p(c)/q(c) x → c

Example Find the limit of the polynomial lim (x 2 + x + 2)/(x + 1) x → 1

The Limit of a Function Involving a Radical Let n be a positive integer. The following limit is valid for c if n is odd and is valid for c>0 if n is even. lim n  x = n  c x → c

The Limit of a Composite Function If f and g are functions such that lim g(x) = L and lim f(x) = f(L), x → c x → L then lim f(g(x)) = f(lim g(x) = f(L) x → c x → c

Example Find the limit each function lim (x 2 + 4)½ x → 0 lim (2x 2 -10) 3/2 x → 3

Functions That Agree at All But One Point Let c be a real number and let f(x) = g(x) for all x ≠ c in an open interval containing c. If the lim of g(x) as x approaches c exists, then the limit of f(x) also exists and then lim f(x) = lim g(x) x → c x → c

Example Find the limit: lim (x 3 - 1)/(x -1) Hint: factor x → 1 lim [(x + 1) ½ - 1]/x Hint: rationalize x → 0

The Squeeze Theorem If h(x) ≤ f(x) ≤ g(x) for all x in an open interval containing c, except possibly at c itself, and if lim h(x) = L = lim g(x) x → c x → c then lim f(x) exists and is equal to L. x → c

Section 3.4 Continuity and One-Sided Limits

Definition of Continuity Continuity at a Point: A function f is continuous at c if the following three conditions are met. 1. f(c) is defined 2. lim f(x) exists x → c 3. lim f(x) = f(c) x → c 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. A function that is continuous on the entire real line (-∞, ∞) is everywhere continuous.

Discontinuity Nonremovable Removable  A discontinuity at c is called nonremovable if f cannot be made continuous by appropriately defining (or redefining f(c)  Example f(x) = 1/x  A discontinuity at c is called removable if f can be made continuous by appropriately defining (or redefining f(c)  Example f(x) = (x 2 -1)/(x -1)

One-Sided Limits and Continuity on a Closed Interval The limit from the right means that x approaches c from values greater than c lim f(x) = L x → c + The limit from the left means that x approaches c from values less than c lim f(x) = L x → c - One-sided limits are useful in taking limits of functions involving radicals. For instance, if n is an even integer, lim n  x = 0 x → 0 +

The Existence of a Limit Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if lim f(x) = L and lim f(x) = L x → c - x → c +

Definition of Continuity on a Closed Interval A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and lim f(x) = f(a) and lim f(x) = f(b) x → a + x → b- The function f is continuous from the right at a and continuous from the left at b

Example Is the function f(x) = (1-x 2 ) ½ continuous? First, determine the domain of f Next use the definition of Continuity on a Closed Interval

Properties of Continuity If b is a real number and f and g are continuous at x = c, then the following functions are also continuous at c. 1. Scalar multiple: bf 2. Sum and difference: f ± g 3. Product: fg 4. Quotient: f/g, if g(c) ≠ 0

Types of Continuous Function at Every Point in their Domain If b is a real number and f and g are continuous at x = c, then the following functions are also continuous at c. 1. Polynomial functions: p(x) = a n x n + a n-1 x n-1 +…. 2. Rational functions: r(x) = p(x)/q(x), q(x) ≠ 0 3. Radical functions: f(x) = n  x

Continuity of a Composite Function If g is continuous at c and f is continuous at g(c), then the composite function given by (f ° g)(x) = f(g(x)) is continuous at c.

Intermediate Value Theorem If f is continuous on the closed interval [a,b] and k is any number between f(a), and f(b), then there is at least one number c in [a,b] such that f(c) = k

Example Use the intermediate value theorem to show the polynomial has a zero in the interval [0,1] f(x) = x 3 + 2x First, is the function continuous on the interval 2. Find f(0) and f(1) then compare relationship to determine if there is a zero in the interval or not

Section 3.5 Infinite Limits

Definition of Vertical Asymptote 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

Vertical Asymptote 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 h(x) = f(x)/g(x) Has a vertical asymptote at x = c

Example Find the vertical asymptotes for each function f(x) = 1/[2(x+1)] f(x) = (x 2 +1)/(x 2 – 1) f(x) = (x 2 – 1)/(x – 2) f(x) = (x 2 + 2x – 8)/ (x 2 – 4)

Properties of Infinite Limits Let c and L be real numbers and let f and g be functions such that lim f(x) = ∞ and lim g(x) = L x → c x → c 1. Sum or difference: lim[f(x) ± g(x)] = ∞ x → c 2. Product:lim[f(x) g(x)] = ∞, L > 0 x → c lim[f(x) g(x)] = -∞, L < 0 x → c 3. Quotient:lim g(x)/f(x) = 0 x → c

Example Find the limit of each function lim (1 + 1/x 2 ) x → 0 lim (x 2 + 1)/1/(x-1)) x → 1 -

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