Analisi Matematica 4 – A.A. 2016/17
Corso di Laurea in Matematica
Esercizi 2
2.1. Let
the following
statements. (a) If f ∈ L+ , the set
(X , M, µ) be a measure space. Prove
1
x : f (x) 6= 0} is σ-finite. (b) If f ∈ L (µ), the set x : f (x) 6= 0} is σ-finite.
2.2. Evaluate the following limits, justifying your answers:
Z k
Z k
x k −2x
x k x/2
1+
e dx
lim
1−
e
dx .
lim
k→+∞ 0
k→+∞ 0
k
k
(Here and in what follows, dx = dm(x) denotes the Lebesgue measure on the real line.)
2.3. Compute the following limits, justifying your calculations:
Z 1
Z +∞
1 + nx
sin(x/n)
dx ,
(ii) lim
dx ,
(i) lim
n→+∞ 0 [1 + x2 ]n
n→+∞ 0
[1 + (x/n)]n
Z +∞
Z +∞
n sin(x/n)
n
(iii) lim
dx
,
(iv)
lim
dx (a ∈ R) .
2
n→+∞ 0
n→+∞ a
x(1 + x )
1 + n2 x2
2.4. Evaluate the following limits, justifying your answers:
√
Z
tanh( x/n)
1
p
(1) lim
dx;
n→+∞ n [k−2 ,+∞) x2 (n + 1)x
√
Z
arctan( 4 x/n)
p √
(2) lim
dx;
n→+∞ [k−1 ,3] x2 x( n + 1)
√
Z
arctan( 4 x/n)
p √
dx;
(3) lim
n→+∞ [k−1 ,+∞) x2 x( n + 1)
Z
log(1 + x/n)
√
(4) lim
dx;
n→+∞ [k−3/2 ,+∞) x3 n + x
Z +∞
1 − 2xn
dx.
(5) lim
n→+∞ 0
1 + 4xn−1 ex−1
2.5. Evaluate the following limits, if they exist, justifying your answers:
Z k
Z +∞
1
1
1
1 −(x−k)2
(i) lim √
(ii) lim 2
e
dx .
√
1 2 dx ,
(x−
)
1
k→+∞
k→+∞ k
x3
k 0
xe k
k
2.6. Determine whether the following equalities hold true
Z +∞
Z +∞
lim
fk (x) dx =
lim fk (x) dx ,
k→+∞ 0
0
k→+∞
where
2
tanh( xk )
1
(i) fk (x) = χ[ 1 ,+∞) (x)
k k
x5
justifying your answers.
2
arctan( xk )
1
(ii) fk (x) = 2 χ[ 1 ,+∞) (x)
,
k
k
x6