Section 1.1_ Limit of a Function - Chapter 1 Section 1.1...

Chapter 1, Section 1.1: Limit of a Function , remixed by Jeff Eldridge from work by Dale Hoffman , is
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62 contemporary calculus
1 . 1 The Limit of a Function
Calculus has been called the study of continuous change, and the limit
is the basic concept that allows us to describe and analyze such change.
An understanding of limits is necessary to understand derivatives,
integrals and other fundamental topics of calculus.
The Idea (Informally)
The limit of a function at a point describes the behavior of the function
when the variable is near—but does not equal—a specified number (see
margin figure). If the values of f ( x ) get closer and closer—as close as
we want—to one number L as we take values of x very close to (but
not equal to) a number c , then
The symbol
→
means “approaches” or
“gets very close to.”
we say: “the limit of f
(
x
)
, as x approaches c , is L ”
and we write: lim
x → c
f
(
x
) =
L
It is very important to note that:
f ( c ) is a single number that describes the behavior (value)
of f at the point x
=
c
while:
lim
x
→
c
f
(
x
)
is a single number that describes the behavior
of f near , but not at the point x = c
If we have a graph of the function f ( x ) near x = c , then it is usually
easy to determine lim
x
→
c
f
(
x
)
.
Example 1 . Use the graph of y
=
f
(
x
)
given in the margin to determine
the following limits:
(a) lim
x → 1
f ( x ) (b) lim
x → 2
f ( x ) (c) lim
x → 3
f ( x ) (d) lim
x → 4
f ( x )