chapter 11

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11.5

area problem

The exact area of a plane region R is given by the limit of the sum of n rectangles as n approaches .

The area problem is to find the area of the region R bounded by the graph of a nonnegative, continuous function f, the x-axis, and the vertical lines x = a and x = b.

limits of summations

begin by applying summation formulas and properties to convert the summation form to rational form. Once in rational form, you can use the techniques from the previous section to find the limit as n .

11.4

define limit of a sequence

Let f be a function of a real variable, such that lim f (x) = L. x If {an} is a sequence such that f (n) = an for every positive integer n, then lim an = L 

if f(x) is a rational function and the limit of f is taken as x approaches infinity or - infinity

When the degree of the numerator is less than the degree of the denominator, the limit is 0 . When the degrees of the numerator and the denominator are equal, the limit is the ratio of the coefficients of the highest-powered terms . When the degree of the numerator is greater than the degree of the denominator, the limit does not exist  .

limits of sequences

For a sequence whose nth term is an, as n increases without bound, if the terms of the sequence get closer and closer to a particular value L, then the sequence is said to converge to L. Otherwise, a sequence that does not converge is said to diverge .

limits at infinity

If f is a function and L1 and L2 are real numbers, the statements lim f (x) = L1 and lim f (x) = L2 denote the limits at infinity. xx The first is read “the limit of f (x) as x approaches   is L1,” and the second is read “the limit of f (x) as x approaches  is L2.”

11.3

difference quotient

the ratio [f(x + h)   f(x)]/h

slope of a graph

To visually approximate the slope of a graph at a point, draw the tangent line to the graph at the point. Then approximate the slope of the tangent line by estimating the change in the value of y for each unit change in x. This ratio approximates the slope of the graph at the point

secant line

a line through the point through the point of tangents and a second point

tangent line

the line that best approximates the slope of the graph at the point

11.2

one sided limit

the limit at c of the function f (x) as x approaches c from either just the left or just the right. A limit from the left is denoted as lim f (x) = L. A limit from the xc right is denoted as lim f (x) = L. x c+

indeterminate form

the fraction 0/0 that results when direct substitution produces 0 in both the numerator and the denominator. It has no meaning as a real number and is called an indeterminate form because it is not possible to determine the limit from the form alone

rationalizing technique

Another way to find the limits of some functions is to first rationalize the numerator. This is called the rationalizing technique , which means multiplying the numerator and denominator by the conjugate of the numerato

validity of the dividing out technique

the fact that if two functions agree at all but a single number c, they must have identical limit behavior at x = c .

11.1

limit of f(x) does not exist under any of these circumstances

1. f (x) approaches a different number from the right side of c than it approaches from the left side of c. 2. f (x) increases or decreases without bound as x approaches c. 3. f (x) oscillates between two fixed values as x approaches c.

properties of limits and direct substitution

1. Scalar multiple: 2. Sum or difference: 3. Product: 4. Quotient: 5. Power:bL L K lim[f(x) g(x)]   LK xc lim f (x) L/K, provided K   0 x cg(x) lim[f(x)]n Ln xc

1. limb xc 2. lim x xc b c 3. limxn   cn xc 4. lim n x xc n   c , for n even and c > 0

limit

If f (x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f (x) as x approaches c is L. This is written as lim f (x) = L.