Definition of Limit,Properties of LimitsDefinition of Limit,Properties of Limits
Section 2.1aSection 2.1a
Let’s start with an exploration…
What are the values of the function given below as
x approaches 0???
First, graph the function in the
window by
Now, look at a table, with
TblStart = –0.3 and Tbl = 0.1
–.3.98507
–.2.99335
–.1.99833
0ERROR
.1.99833
.2.99335
.3.99507
Let’s start with an exploration…
What are the values of the function given below as
x approaches 0???
–.3.98507
–.2.99335
–.1.99833
0ERROR
.1.99833
.2.99335
.3.99507
What do these steps suggest?
Note: We cannot simply substitute x = 0Note: We cannot simply substitute x = 0
into the function, because we’d be dividinginto the function, because we’d be dividing
by zero…………we need another method…by zero…………we need another method…
Definition: LimitDefinition: Limit
Let c and L be real numbers. The function f has limit L
as x approaches c if, given any positive number ,
there is a positive number such that for all x,
We write
which is read, “the limit of f of x as x approaches c equals L.”
The notation means that the values of f(x) of the function f
approach or equal L as the values of x approach (but do not
equal ) the number c…
Definition: LimitDefinition: Limit
Let c and L be real numbers. The function f has limit L
as x approaches c if, given any positive number ,
there is a positive number such that for all x,
We write
As suggested in our opening exploration:
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Definition: LimitDefinition: Limit
Let c and L be real numbers. The function f has limit L
as x approaches c if, given any positive number ,
there is a positive number such that for all x,
We write
Find each of the following limits:
Properties of LimitsProperties of Limits
If L, M, c, and k are real numbers and
and
, then
1.Sum Rule – The limit of the sum of two functions is the sum of
their limits:
2. Difference Rule – The limit of the difference of two functions
is the difference of their limits:
Properties of LimitsProperties of Limits
If L, M, c, and k are real numbers and
and
, then
3. Product Rule – The limit of a product of two functions is the
product of their limits:
4. Constant Multiple Rule – The limit of a constant times a
function is the constant times the limit of the function:
Properties of LimitsProperties of Limits
If L, M, c, and k are real numbers and
and
, then
5. Quotient Rule – The limit of a quotient of two functions is the
quotient of their limits, provided the limit of the denominator is
not zero:
Properties of LimitsProperties of Limits
If L, M, c, and k are real numbers and
and
, then
6. Power Rule – If r and s are integers, , then
provided that is a real number.
The limit of a rational power of a function is that power of the
limit of the function, provided the latter is a real number.
Guided PracticeGuided Practice
Find each of the following limits.
(a)
(b)
NoticeNotice
anything?anything?
Theorem:Theorem:
Polynomial and Rational FunctionsPolynomial and Rational Functions
1. If
is any
polynomial function and c is any real number, then
2. If
are polynomials and c is any real
number, then
and
provided that
Guided PracticeGuided Practice
Find each of the following limits.
(a)
(b)
Guided PracticeGuided Practice
Explain why you cannot use substitution to determine the given
limits. Find the limit if it exists.
Cannot use substitution b/c the expression
is not defined at x = 0.
Since becomes arbitrarily large as x approaches
0 from either side, there is no (finite) limit.
Can we support this reasoning graphically???
Guided PracticeGuided Practice
Explain why you cannot use substitution to determine the given
limits. Find the limit if it exists.
Cannot use substitution b/c the
expression is not defined at x = 0.
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Guided PracticeGuided Practice
Explain why you cannot use substitution to determine the given
limits. Find the limit if it exists.
Cannot use substitution b/c the
expression is not defined at x = 0.
Support graphically???
Guided PracticeGuided Practice
Determine the given limits algebraically. Support graphically.
Guided PracticeGuided Practice
Determine the given limits algebraically. Support graphically.
Guided PracticeGuided Practice
Determine the given limits algebraically. Support graphically.
