Question

Calculati:
\lim_{n \to \infty} ([n/4])/n
Unde [n/4] reprezinta partea intreaga a lui n/4

Answer 1

[tex] \frac{n}{4} -1 \leq [ \frac{n}{4}] \leq \frac{xn}{4} \lim_{n \to \infty} ( \frac{n}{4}-1) \leq \lim_{n \to \infty} [ \frac{n}{4}] \leq \lim_{n \to \infty} \leq \lim_{n \to \infty} \frac{n}{4} [/tex]  |:n  (imparti fiecare chestie la n )

$\lim_{n \to \infty} \frac{n-4}{4n} \leq \lim_{n \to \infty} [\frac{n}{4}]* \frac{1}{n} \leq \lim_{n \to \infty} \frac{n}{4n}$


In cazul primei limite, observi ca ai infinit/infinit, asa ca aplici L'Hospital si obtii 1/4.
In cazul ultimei limite, se reduce n cu n si ramane 1/4.
Daca prima si ultima tind la 1/4, inseamna ca si [n/4]/n tinde tot la 1/4.