B - Prírodovedecká fakulta - Univerzita Komenského

t.
a.
ln x
lim
x→∞
x
Riešenie.
( 11ϕ
)
( 5ϕ
)
u.
x
lim
x→0 tanh x
2 ( 5ϕ
)
=
( 11 ) 5
tan
11cos
= lim
→−π tan
ϕ →−π
5cos
ϕ
lim
ϕ
2
x
e − 1
x
b. lim = lim e = 1
x→0
x x→0
− x
− x
− x
e − e e
c. lim = lim = lim = 0
x→∞
2
x x→∞
2x
x→∞
2
arcsin x 1
d. lim = lim = 1
x→0
x x→0
2
1 − x
sin x
e. lim = lim cos x = 1
x→0
x x→0
f.
g.
( arctanϕ
− π / 2)
2 ( 1 + ϕ )
11
1
= lim
= 0
→∞ 3
ϕ
ϕ →∞
3ϕ
lim
ϕ
2
lim
cos
cos
( 3ϕ
)
( 9ϕ
)
3sin
lim
π / 2 9sin
=
ϕ →π / 2
ϕ → ϕ
ln x 1
h. lim = lim = 0
x→∞ x x→∞
x
e xe
i.
j.
k.
ϕ
( ϕ)
( 5ϕ
)
2
cotg 5sin
lim = lim
→π / 2
2
cotg ϕ →π
/ 2 sin ϕ
lim
x −1
1−
x
−
= lim
x→1
2 x→1
x
( 3ϕ
)
( − 8ϕ
)
2 ( 1−
x )
( 3ϕ
) 1
= −
( 9 ) 3
( 5ϕ
)
= 5
3
= 0
( 3ϕ
) 3
( 8 ) 8
sin
3cos
lim = lim =
ϕ π sin
ϕ →π
− 8cos
→ ϕ
sinh x
l. lim = lim cosh x = 1
x→0
x x→0
Lekcia 8 – Derivácie elementárnych funkcií
ln
lim
x
2
v.
x→∞ 2
x
cosϕ
+ 1 − sinϕ
m. lim = lim = 0
ϕ →π sinϕ
ϕ →π
cosϕ
sinh
x
2
n. lim lim(
2sinh
x.
cosh x)
= 0
o.
x→0
lim
x→∞
1+
x
x
2
x
2
=
x→0
= lim
x→∞
2
1
2 ( 1+
x )
3
= 0
2
sinh x 2sinh
x.
cosh x
p. lim = lim
= 1
x→0
2
cosh x −1
x→0
2sinh
x.
cosh x
sinh x cosh x
q. lim = lim = 1
x→0
sin x x→0
cos x
−1 −
x x→0
x
r. lim = − lim(
2xe
) = 0
x→0
e
2
− x
x
s. lim = lim x = 0
x→0
ln x x→0
ln x 1
t. lim = lim = 0
x→∞
x x→∞
x
x
2
u. lim = lim cosh x = 1
x→0
tanh x x→0
2
ln x ln x 1
v. lim lim = lim = 0
x→∞
2 →∞
2 →∞
2
x x x x 2x
2
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