(Math 250) Chapter 1: Limits And Their Properties

Front 1

slope of the secant line

Back 1

Front 2

The limit of (f):

Back 2

Front 3

common types of behavior associated with Nonexisitance of a Limit:

Back 3

1. f(x) approaches a different number form the right side of c than it approaches from the left side

2. f(x) increases or decreases without bound as x approaches c.

3. f(x) oscillates between two fixed values as x approaches c.

Front 4

Definition of Limit:

Back 4

Front 5

Back 5

b

Front 6

Back 6

c

Front 7

Back 7

Front 8

Back 8

bL

Front 9

Back 9

Front 10

Back 10

LK

Front 11

Back 11

Front 12

Back 12

Front 13

If p is a polynomial function and c is a real number, then

Back 13

p(c)

Front 14

If r is a rational function given by r(x) = p(x)/q(x) and c is a real number such that a(c) can not equal 0, then

Back 14

Front 15

Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c>0 if n is even.

Back 15

Front 16

If f and g are functions such that the limit of g(x)= L as x approaches c and the limit of f(x) = f(L) as x approaches L, then

Back 16

Front 17

Back 17

sin c

Front 18

Back 18

cos c

Front 19

Back 19

tan c

Front 20

Back 20

sec c

Front 21

Back 21

csc c

Front 22

functions that agree at all but one point

let c be a real number and let f(x) = g(x) for all x can not equal c in an open interval containing c. if the limit of g(x) as x approaches c exists, then the limit of c(x) also exisist and

Back 22

Front 23

strategy for finding limits

Back 23

Front 24

The squeeze theorem

Back 24

Front 25

Back 25

1

Front 26

Back 26

0

Front 27

Definition of Continuity

Back 27

Front 28

The Existence of a limit

Back 28

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

Front 29

Definition of Continuity on a closed interval

Back 29

Front 30

The Properties of Continuity

Back 30

if b is a real number aned f and g are continuous at x=c, then the following functions are also continouous at c.

Front 31

Continuity of a Composite function

Back 31

Front 32

The Intermediate Value Theorem

Back 32

If f is continuous on the colsed 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

Front 33

Definition of Infinite Limits

Back 33

Front 34

Definition of Vertical Asymptote:

Back 34

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.

Front 35

Vertical asymptotes:

Back 35

let f and g be continuous on an open interval containing c. If f(c) does not equal 0, g(c) does not equal 0, and there exists an open interval containing c such that g(x) does not equal 0 for all x does not equal 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

Front 36

Properties of Infinite Limits

Back 36