Download 7.7 Indeterminate Forms and LGÇÖHopitalGÇÖs Rule

Quiz 7-3 to 7-5
Trig Integrals
Trig Substitution (notecard)
Partial Fractions
True/False
1) A limit describes how a function moves as
x moves toward a certain point.
2) A limit is a number or point past which a
function cannot go.
3) A limit is a number that the y-values of a
function can be made arbitrarily close to
by restricting x-values.
True/False
4) A limit is a number or point the functions
gets close to but never reaches.
5) A limit is an approximation that can be
made as accurate as you wish.
6) A limit is the value reached by plugging in
numbers closer and closer to a given
number.
Indeterminate Forms and
L’Hopital’s Rule
Section 7.7 AP Calc
Find the limit:
3x  12
lim
x 2
x2
2
4x  7
lim 2
x  5 x  3
2
Thm 7.3 Extended Mean Value Theorem
If f and g are differentiable on an open
interval (a,b) and continuous on [a,b] such
that g’(x)≠0 for any x in (a,b), then there
exists a point c in (a,b) such that
f ' (c) f (b)  f (a)

g ' (c) g (b)  g (a)
Thm 7.4 L’Hopital’s Rule
Let f and g be functions that are
differentiable on an open interval (a,b)
containing c, except possibly at c itself.
Assume g’(x)≠0 for all x in (a,b), except
possibly at c itself. If the limit of f(x)/g(x)
as x approaches c produces the
indeterminate form 0/0, then
f ( x)
f ' ( x)
lim
 lim
x c g ( x )
x c g ' ( x )
f ( x)
f ' ( x)
lim
 lim
x c g ( x )
x c g ' ( x )
provided the limit on the right exists (or is
infinite). This result also applies if the limit
of f(x)/g(x) as a approaches c produces
any one of the indeterminate forms ∞/∞,
(- ∞)/ ∞, ∞/(- ∞), or (- ∞)/(- ∞).
Determine the limits:
e  (1  x)
A) lim
x 0
x
x
2x 1
B) lim
x  4 x 2  x
ln
x
C) lim
x  x 2
4
x
D) lim
x  e 2 x
3
Other indeterminate forms:

0  ,1 ,  ,0 ,   
0
0
Find the limit:
 1
E) lim 1  
x 
 x
x
F) lim x 3 cot x

x 0
x 
 8
G) lim  2


x2  x  4
x2
Use Graphing Calculator to check limits
Notice:
x
x
e
e
lim
 lim
x 0 x
x 0 1