11.1

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.

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

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. 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.

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

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

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

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+

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

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

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

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

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.”

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 .

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  .

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


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 .

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.

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