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L’Hôpital’s Rule
Let f and g be differenti able, such that
a) as x  a, either
i) f  x   0 and g  x   0; or
ii) f  x    and g  x   ;
f  x 
b) lim
exists.
xa g  x 
Then
f x
f  x 
lim
 lim
.
x a g  x 
x a g  x 
What is a sequence?
• An infinite, ordered list of numbers.
{1, 4, 9, 16, 25, …}
{1, 1/2, 1/3, 1/4, 1/5, …}
{1, 0, 1, 0, 1, 0, –1, 0, …}
What is a sequence?
• A real-valued function defined for positive
(or non-negative) integer inputs.
{an}, where an= n2 for n = 1, 2, 3, …
{ak}, where ak= 1/k for k = 1, 2, 3, …
{aj}, where aj= cos((j-1)/2) for j = 1, 2, 3, …
Notation
• Implicit Form
{a1, a2, a3, …}
• Explicit Forms
an 
a 

n 1
an n1
Explicit to Implicit

1. Convert the sequence
2.
1
 n  to
 2 0
implicit form.
2x 1
Given the function f x   3 , write the
x

implicit form of the sequence  f nn1.
Implicit to Explicit
1. Write the sequence
form.
2. Write the sequence
explicit form.
 1 1 1

1
,
,
,
,


 in
 3 9 27 
explicit
 1 1 1 1

 , , , , in
 2 4 8 16 
The Fibonacci Sequence
• Defined by the rules:
F1 = 1
F2 = 1
Fn+2 = Fn + Fn+1
• Implicit Form:
{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …}
• Fibonacci Numbers in Nature
The Big Question
• Once again, it’s this: convergence or
divergence?
– Let {ak} be a sequence and L a real number. If
we can make ak as close to L as we like by
making k sufficiently large, the sequence is said
to converge to L.
lim ak  L or ak   L
k 
– Otherwise, the sequence diverges.
Rigorous Definition
If, for  > 0, there is an integer N such that
ak  L    k  N
then the sequence {ak} is said to converge to the real
number L (i.e., {ak} has the limit L).
Convergence Theorem
Let f be a function defined for x  1. If lim f  x   L
x 
and ak = f (k) for all k  1, then lim ak  L.
k 
Algebra with Limits
If lim an  A and lim bn  B then
n 
n 
1) lim can  cA
n 
2) lim an  bn  A  B
n 
3) lim an  bn  A  B
n 
4) lim anbn  AB
n 
an A
5) lim
 , provided B  0.
n  b
B
n
The Squeeze Theorem
Suppose that
ak  bk  ck for all k  1
and that
lim ak  lim ck  L.
k 
k 
Then
lim bk  L.
k 