L’Hospital’s Rule for Form
Suppose that f and g are differentiable functions on an open interval
containing x = a, except possibly at x = a, and that and
.
If
has a finite limit, or if this limit is
or
,
then
.
Moreover, this statement is also true
in
the case of a limit as
or
as
In the following examples, we will use the following three-step process:
Step
1. Check that the limit of
is an indeterminate form of type
.
If it is not, then L’Hospital’s
Rule cannot be used.
Step 2. Differentiate f and g separately. [Note: Do not differentiate using the quotient rule!]
Step
3. Find the limit of
.
If this limit is finite,
,
or
,
then it is equal to the limit of
.
If the limit is an indeterminate form
of type
,
then simplify
algebraically and apply L’Hospital’s
Rule again.
II.
Indeterminate Form of the Type
We have previously studied limits with the indeterminate form as shown in the following example:
Example:
However, we could use another version of L’Hospital’s Rule.
L’Hospital’s Rule for Form
Suppose that f and g are differentiable functions on an open interval
containing
x = a,
except possibly at x
= a, and
that
and
.
If
has a finite limit, or if this limit is
or
, then . Moreover, this statement is also true
in the case of a limit as or as
III.
Indeterminate Form of the Type
Indeterminate forms of the type can sometimes be evaluated by rewriting the product as a quotient, and then applying L’Hospital’s Rule for the indeterminate
forms of type or .
Example:
IV.
Indeterminate Form of the Type
A limit problem that leads to one of the expressions
,
,
,
is called an indeterminate form of type . Such limits are indeterminate because the two terms exert conflicting influences on the expression; one pushes it in the positive direction and the other pushes it in the negative direction. However, limits problems that lead to one the expressions
,
,
,
are
not indeterminate, since the two terms work together (the first two
produce a limit of
and the last two produce a limit of
).
Indeterminate forms of the type
can sometimes be evaluated by combining the terms and manipulating
the result to produce an indeterminate form of type
or
.
Example:
V.
Indeterminate Forms of the Types
Limits
of the form
frequently give rise to indeterminate forms of the types
.
These indeterminate forms can sometimes be evaluated as follows:
The
limit on the right hand side of the equation will usually be an
indeterminate limit of the type
.
Evaluate this limit using the technique previously described.
Assume that
=
L.
(4)
Finally,
.
Example:
Find
.
This
is an indeterminate form of the type
.
Let
.
0.
Thus,
.
Taylor’s formula
Suppose we’re working with a function f(x)
that is continuous and has n+1
continuous derivatives on an interval about x
= 0. We can approximate f
near 0 by
a polynomial
of degree n:
• For n = 0,
the best constant approximation near 0
is
which matches f
at 0.
• For n = 1,
the best linear approximation near 0
is
.
Note that
matches f
at 0 and
matches
at 0.
• For n = 2,
the best quadratic approximation near 0
is
.
Note that
,
,
and
match
,
, and
,
respectively, at 0.
Continuing this process,
.
This is the Taylor polynomial of degree n about 0 (also called the Maclaurin series of degree n). More generally, if f has n+1 continuous derivatives at x = a, the Taylor series of degree n about a is
.
This formula approximates f (x) near a. Taylor’s Theorem gives bounds for the error in this approximation:
Taylor’s Theorem:
Suppose f has n+1 continuous derivatives on an open interval containing a. Then for each x in the interval,
,
where the error term
satisfies
for some c between a and x.
This form for the error , derived in 1797 by Joseph Lagrange, is called the Lagrange formula for the remainder. The infinite Taylor series converges to f ,
,
if
and only if
.