FINITE MATH
Section 8.1 – Limits and Continuity
Simple definition of Calculus – The study of rates of change.
We will use calculus to analyze rates of inflation, rates of learning, rates of population
growth and rates of natural resource consumption.
LIMITS
Limit (everyday definition) – the ultimate behavior of something. Ex. The “limit of one’s
endurance” or the “limit of one’s patience.”
x  3 is read “x approaches 3” and means that x takes values arbitrarily close to 3
without ever reaching 3.
Example: The numbers 2.9, 2.99, 2.999, . . . approach 3 (from the left), and the numbers
3.1, 3.01, 3.001, . . . approach 3 (from the right). Note that x takes values closer and
closer to 3 but never equals 3.
This looks like the following on a number line:
Applying this idea to functions, given a function f(x), if x approaching 3 causes the
function to take values arbitrarily close to a y-value (for ex. Y = 10), then we call 10 the
limit of the function, and write
lim f ( x)  10
x 3
IF the limit exists (sometimes it doesn’t), it must be a single number.
DEFINITION (informal):
Limit of a Function: lim f ( x ) is the number that f ( x) approaches as x approaches
x c
c ( x  c ).
FINDING LIMITS
1. Finding a Limit Using Tables
Example 1: Use tables to find lim(2 x  4) .
x 3
We choose x-values approaching 3, and calculate resulting values of
f ( x)  2 x  4 to create the following tables.
Table for x<3
x
2x + 4
2.9
9.8
2.99
9.98
2.999
9.998
Table for x>3
x
2x + 4
3.1
3.01
3.001
10.2
10.02
10.002
Looking at the tables we see that lim(2 x  4)  10
x 3
The graph of f(x) = 2x + 4 is shown below. Again we can see that as x
approaches 3 (on the x-axis), f(x) approaches 10 (on the y-axis).
In this example, the correct limit could have been found by using “direct
substitution,” or simply evaluating the function at x  3 as follows.
f (3)  2  3  4  10
However, this technique of direct substitution is not always possible as the next
example shows.
Example 2: Find lim(1  x)1 x correct to three decimal places.
x 0
Attempting to evaluate the limit using direct substitution would result in division
by zero which is undefined. Therefore, we will use tables to evaluate the limit.
The function values in the following tables were found using a calculator:
From the tables we see the following:
lim(1  x)1 x  2.718
x 0
RULES OF LIMITS
For any constants a and c , and any positive integer n :
1. lim a  a
x c
2. lim x n  c n
x c
3. lim n x  n c (c > 0 if n is even)
x c
4. If lim f ( x ) and lim g ( x) both exist, then
x c
x c
a. lim  f ( x)  g ( x)  lim f ( x)  lim g ( x)
x c
x c
x c
x c
x c
x c
b. lim  f ( x)  g ( x)  lim f ( x)  lim g ( x)
c. lim  f ( x)  g ( x)  lim f ( x)   lim g ( x) 
x c
 xc
  xc

lim
f
(
x
)
f ( x ) x c

d. lim
(if lim g ( x)  0)
x c
x c g ( x )
lim g ( x)
x c
Summary of 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) (Limit evaluated by direct substitution)
x c
Examples:
a.
lim x 
x 4
b. lim
x 6
x2

x3
c. lim(2 x2  4 x  1) 
x 3
*If direct substitution into a quotient gives the undefined expression
0
, factoring,
0
simplifying, and then using direct substitution may help.
Examples:
x2  1
1. Find lim
x 1 x  1
2 x 2  10 x
x 5
x 5
2. Find lim
ONE-SIDED LIMITS
 In a one-sided limit, the variable approaches the number from one side only.
 Left and Right Limits
- lim f ( x) means the limit of f ( x) as x  c but with x < c
x c

lim f ( x) means the limit of f ( x ) as x  c but with x > c
x c
lim f ( x)  L if and only if both one-sided limits lim f ( x) and lim f ( x) exist
x c
x c
x c
and equal the same number L.
Example:
 x  1, x  3
For the piecewise linear function f ( x)  
graph below, find the following
8  2 x, x 3
limits or state that they do not exist.
a. lim f ( x) 
x 3
b. lim f ( x) 
x 3
c. lim f ( x) 
x 3
INFINITE LIMITS
 ∞ and -∞ are used to indicate that the values of a function become arbitrarily large
or arbitrarily small.
 The graphs in the examples below show some of examples of how the values of a
function become arbitrarily large or arbitrarily small. The dashed lines on the
graphs, where the function approaches ∞ or -∞, are called vertical asymptotes.
 Examples: For the functions graphed below, use the limit notation with ∞ and -∞
to describe their behaviors as x approaches the vertical asymptote from the left,
from the right, and from both sides.
a.
lim f ( x) 
lim f ( x) 
x  2
x  2
lim f ( x) 
x2
b.
lim f ( x) 
lim f ( x) 
x 3
x 3
lim f ( x) 
x 3
WARNING! To say that a limit exists means that the limit is a number, and since ∞ and ∞ are not numbers, a statement such as lim f ( x)   means that the limit does not exist.
x c
The limit statement lim f ( x)   goes further to explain why the limit does not exist: the
x c
function values become too large to approach any limit.
Examples: a. Find lim( x2  xh  h2 )
h0
b. Find
lim(3x2  5xh  1)
h0
CONTINUITY
A function is continuous at c if the following three conditions hold:
1. f (c ) is defined
2. lim f ( x ) exists
x c
3. lim f ( x ) = f (c )
x c
f is discontinuous at c if one or more of these conditions fails to hold.
Finding Discontinuities From a Graph
Example 1 Given the graph of f(x), shown below, determine if f(x) is continuous at
,
, and
. If it is not continuous, indicate the first of the three conditions in the definition of continuity
(on the previous page) that is violated.
Jump discontinuity. Jump discontinuities occur where the graph has a break in it is as this graph does at
x = -2.
Removable discontinuity. Removable discontinuities are those where there is a hole in the graph as there
is at x = 3.
Examples of Types of Functions That Are Always Continuous:
- Linear and Quadratic functions – since graphs are straight lines and parabolas,
with no holes or jumps, they are continuous.
- Exponential and Logarithmic functions – graphs are continuous with no holes or
jumps.
Combining continuous functions:
If functions f and g are continuous at c, then the following are also continuous at c :
1. f  g
2. a  f (for any constant a )
3. f  g
4. f g [if g (c)  0]
5. f ( g ( x))
[for f continuous at g (c) ]
Determining Continuity
Examples – Determine whether each function is continuous.
a.
f ( x)  x 3  3x 2  x  3
1
b. f ( x) 
2
 x  1
c.
f ( x)  e x
2
1