4.4 Indeterminate Forms and l’Hospital’s Rule
12)
lim sin 4x
x→0 tan 5x
indeterminate form
0
0
x2
1−cos
x
x→0
indeterminate form
0
0
2x
x→0 sin x
indeterminate form
0
0
indeterminate form
0
0
indeterminate form
∞
∞
Indeterminate form
∞
∞
indeterminate form
0
0
indeterminate form
0
0
4 cos 4x
2
x→0 5 sec 5x
= lim
14)
=
4
5
lim
= lim
2
x→0 cos x
= lim
18)
lim ln x
x→1 sin πx
1
x→1 x
= lim
20)
=2
lim
x→∞
·
1
π cos πx
= − π1
ln(ln x)
x
1
1
x→∞ ln x x
·
= lim
1
1
x
x→∞ ln x
= lim
= lim
x→∞
22)
lim
t→0
1
1
x
= lim x = ∞
x→∞
8t −5t
t
= lim 8t ln 8 − 5t ln 5
t→0
= ln 8 − ln 5 = ln 85
28)
lim x−sin x
x→0 x−tan x
1−cos x
2
x→0 1−sec x
= lim
(1−cos x)
2
x→0 cos x−1
= lim
· cos2 x
2
cos x
1
= lim − cos
x+1 = − 2
x→0
1
34)
x
lim
−1
x→0 tan (4x)
lim
x→0
42)
lim
1
√
x→∞
Indeterminate form
∞
∞
x
xe− 2
= lim
p
x→∞
46)
0
0
1
4
=
4
1+(4x)2
indeterminate form
x
ex
=
q
lim xx
x→∞ e
=
q
lim 1x
x→∞ e
=0
lim x tan( x1 )
indeterminate form ∞ · 0
x→∞
= lim
1
tan( x
)
1
x
x→∞
1
)
−x−2 sec2 ( x
−2
−x
x→∞
= lim
= lim sec2 ( x1 ) = 1
x→∞
62)
1
lim (ex + x) x
indeterminate form ∞0
x→∞
= lim e
ln(ex +x)
x
x→∞
= elimx→∞
= elimx→∞
= elimx→∞
= elimx→∞
ln(ex +1)
x
ex +1
ex +x
ex
ex +1
ex
ex
=e
2
indeterminate form
∞
∞
indeterminate form
∞
∞
indeterminate form
∞
∞