ESERCIZI
C ONTENTS
1.
2.
3.
(1)
(2)
(3)
(4)
(5)
Determinare il dominio delle seguenti funzioni
Calcolare i seguenti limiti
Determinare i valori del parametro affinché le funzioni siano continue
1. D ETERMINARE IL DOMINIO DELLE SEGUENTI FUNZIONI
x+1
.
f (x) = log
x − 1 1
f (x) = log
.
x(x − 1)
log(x + 2)
f (x) =
.
3 − x2
log(x + 3)
.
f (x) =
5 − x2
log(log x) .
2. C ALCOLARE I SEGUENTI LIMITI
(1)
(2)
(3)
(4)
(5)
(6)
sin((x − 1)2 )
lim
.
x→1
x−1
2x
cos(e + 1)
lim
.
x→+∞
x3
x−2
x
lim
.
x−2
x→2+
sin(x2 − 1)
lim
.
x→1
x−1
cos((2x + 1)e3x )
lim
.
x→+∞
x3/2
sin x
lim+ x
.
x→0
lim (1 − cos(1/x))(1 + sin(ex )).
1
x sin(2x)
(8) lim 1 −
.
x→0
2
√
2
(sin x)
(9) lim
.
2x
x→0+
1
(10) lim+ (cos x) log x .
(7)
x→+∞
x→0
(11)
(12)
lim (x − 1) log(x2 − 1).
x→1+
lim sin(1/x)(1 + cos(ex )).
x→+∞
1
(13) lim (1 − 2x) sin(x/2) .
x→0
sin(2x − π)
.
3x
sin(log(2x))
(15) lim
.
x→+∞
log(x3 )
(16) lim+ (tan x)tan x .
(14) lim
x→0
x→0
1
1
1
2
2
ESERCIZI
sin(−2x)
.
3x
arctan(x3 )
(18) lim
.
x→+∞
x3
sin x
(19) lim (2x)
.
(17) lim
x→0
x→0+
log x
√ .
x
sin(log x)
(21) lim
.
x→+∞
x
h x i log1 x
.
(22) lim sin
3
x→0+
(20)
lim
x→0+
3. D ETERMINARE I VALORI DEL PARAMETRO AFFINCHÉ LE FUNZIONI SIANO CONTINUE
(
c + log(x + e), x > 0,
(1) f (x) =
2
e−x ,
x < 0.
(
e−x
log( c ), x < 0,
(2) f (x) =
2
e−x ,
x > 0.
( 2x −x
e −e
, x > 0,
x
(3) f (x) =
−|x| + c, x < 0.
(
arcsin(1 − x2 ), |x| < 1,
(4) f (x) =
log(2x2 − c),
|x| > 1.
(
2
x log x,
x > 0,
(5) f (x) =
2
(x − c) + x + 9, x < 0.
( 1
x < −1,
e x+1 ,
(6) f (x) =
log(x + 1 + c), x > −1.

n o 2 - IRMA

Hs9 Exponentiële en logaritmische functies