Question

.......................................................................................................................

Answer 1

$\lim\limits_{x\to\infty} [f(x)^{g(x)}] = [\lim\limits_{x\to\infty}f(x)]^{\lim\limits_{x\to\infty}g(x)}\\\text{- cand }f(x) > 0\\ \\l = \lim\limits_{n\to\infty} \Bigg(\dfrac{n}{7n+3}\Bigg)^n\\ \\ l = \Big[\lim\limits_{n\to\infty} \dfrac{n}{7n+3} \Big]^{\lim\limits_{n\to\infty} n} \\ \\ l = \Big(\dfrac{1}{7}\Big)^{\infty}\\ \\ \Rightarrow \boxed{l = 0}$

Answer 2

Răspuns:

0

Explicație pas cu pas:

Aplicam formula $a^{log_a~b}=b$, dar si proprietatea de la logaritmi care ne spune ca $log_a b^c=c*log_ab$.

Totodata, stim si ca $lim_{n \to \infty} e^{a_n}=e^{lim_{n \to \infty} a_n}$, aₙ fiind un sir oarecare.

$L=\lim_{n \to \infty} (\frac{n}{7n+3})^n= \lim_{n \to \infty} e^{ln(\frac{n}{7n+3})^n}= \lim_{n \to \infty} e^{nln(\frac{n}{7n+3})}=e^{ \lim_{n \to \infty} nln(\frac{n}{7n+3})}$

Calculam separat limita ramasa la exponent.

Aplicam, totodata, si formula: $lim_{n \to \infty} a_n*b_n=lim_{n \to \infty} a_n*lim_{n \to \infty}b_n$, cand sirurile sunt finite (au numar finit de termeni).

${ \lim_{n \to \infty} nln(\frac{n}{7n+3})}=lim_{n \to \infty}n*lim_{n \to \infty}ln(\frac{n}{7n+3})=\infty*lim_{n \to \infty} ln(\frac{n}{n(7+\frac{3}{n})})=\infty*ln(\frac{1}{7})=\infty*(-ln7)=-\infty$

Si ne intoarcem la limita noastra:

$L=lim_{n \to \infty} (\frac{n}{7n+3})^n=e^{ \lim_{n \to \infty} nln(\frac{n}{7n+3})}=e^{-\infty}=\frac{1}{e^{\infty}}=0$