
8-16 Calculus – Differentiation and Integration
=
=
−
→
→
lim
log cos
tan
lim
( / cos )( sin )
tan se
x
x
x
x
x x
x
0
2
0
1
2
[Form 0/0]
cc
lim
tan
tan sec
lim .
2
0
2
0
2
2
1
2
1
2
x
x
x x
x
x
x
=
−
=
−
= −
→
→
sec
∴ = −
=
−
log /
.
/
y e
1 2
1 2
Example 8.24 Evaluate lim .
x
x
x
®0
Solution: Let
y x
x
x
=
→
lim
0
0
0[Form ]
Then,
log lim log
lim
log
/
/
lim
y x x
x
x
x
x
x
= ⋅∞
= ∞ ∞
=
→
→
→
0
0
0
0
1
1
[Form ]
[Form ]
//
/
lim( ) .
.
x
x
x
y e
x
−
= − =
∴ = =
→
1
0
1
2
0
0
Example 8.25 Evaluate lim .
/
x
x
x
→∞
1
Solution: Let
y x
x
x
= ∞
→∞
lim ]
/1 0
[Form
Then,
log ln
lim
lim ln
lim
ln
y
x
x
x
x
x
x
x
x
x
=
( )
= ⋅∞
= ∞
→∞
→∞
→∞
1 1
0[Form ]
[Form
// lim
/
.
log .
∞ = =
∴ =
→∞
]
x
x
y
1
1
0
0
Thus,
y x e
x
x
= = =
→∞
lim .
/1 0
1