
Indeterminate Forms 8-11
Example 8.14 Evaluate lim .
x
n x
x e
→∞
−
Solution: We have
lim lim /
lim lim
x
n x
x
n
x
x
n
x
x
n
x e
x
e
nx
e
n
x
→∞
−
→∞
→∞
−
→∞
−
= ∞ ∞
= =
[Form ]
1 1
ee
n n
x
e
n n n
x
x
n
x
x
[Form ]
[Form ]
∞ ∞
= − ∞ ∞
= − −
→∞
−
/
( ) lim /
( )( ) lim
1
1 2
2
→→∞
−
→∞
−
→∞
−
∞ ∞
= − − =
x
e
n n n
x
e
n e
n
x
x
n n
x
x
x
3
1 2 1
[Form ]/
( )( ) lim ! lim
== 0.
8.5 L’ HOSPITAL’S RULE: FORM 3: 0 × ¥
Let f(x) and g(x) be two functions such that
lim ( )
x a
f x
→
= 0 and lim ( ) ,
x a
g x
→
= ∞ and lim ( ) ( ) .
x a
f x g x
→
= ⋅∞0
Then we transform this form into either of the forms 0/0 and ¥/¥, as we have
lim ( ) ( ) lim
( )
/ ( )
lim
( )
/ (
x a x a
x a
f x g x
g x
g x
f
→ →
→
=
[ ]
=
1
1
[Form 0/0]
xx)
[ ]
∞ ∞[Form / ]
so that ...