Oblicz granicę ciągu...- Zadanie 1: MATeMAtyka 3. Zakres podstawowy i rozszerzony

Rozwiązanie

Jeśli

$n \to \infty lim a_{n} = a \neq = 0$

to zachodzi

$Gdy n \to \infty lim b_{n} = \infty$	$Gdy n \to \infty lim b_{n} = - \infty$
$n \to \infty lim (a_{n} \cdot b_{n}) = \infty, dla a > 0$	$n \to \infty lim (a_{n} \cdot b_{n}) = - \infty, dla a > 0$
$n \to \infty lim (a_{n} \cdot b_{n}) = - \infty, dla a < 0$	$n \to \infty lim (a_{n} \cdot b_{n}) = \infty, dla a < 0$

Oraz

$n \to \infty lim a_{n} = \infty, n \to \infty lim b_{n} = \infty \Rightarrow n \to \infty lim (a_{n} \cdot b_{n}) = \infty$

$n \to \infty lim a_{n} = - \infty, n \to \infty lim b_{n} = \infty \Rightarrow n \to \infty lim (a_{n} \cdot b_{n}) = - \infty$

$n \to \infty lim a_{n} = \infty, n \to \infty lim b_{n} = - \infty \Rightarrow n \to \infty lim (a_{n} \cdot b_{n}) = - \infty$

$n \to \infty lim a_{n} = - \infty, n \to \infty lim b_{n} = - \infty \Rightarrow n \to \infty lim (a_{n} \cdot b_{n}) = \infty$

$a) n \to \infty lim (10 n - 4 n^{3}) = n \to \infty lim n (10 - 4 n^{2}) = \infty \cdot (10 - 4 \cdot \infty) = \infty \cdot (- \infty) = - \infty$

$b) n \to \infty lim (3 n^{5} - 20 n) = n \to \infty lim n (3 n^{4} - 20) = \infty \cdot (\infty - 20) = \infty \cdot \infty = \infty$

$c) n \to \infty lim (100 n^{3} - 5 n^{6}) = n \to \infty lim n^{3} (100 - 5 n^{3}) = \infty \cdot (100 - 5 \cdot \infty) = \infty \cdot (- \infty) = - \infty$

$d) n \to \infty lim (5 n^{3} - n^{2} - 9) = n \to \infty lim n^{3} (5 - \frac{1}{n} - \frac{9}{n ^{3}}) = \infty \cdot (5 - 0 - 0) = \infty \cdot 5 = \infty$

$e) n \to \infty lim (- 3 n^{4} + 9 n^{2} + 8) = n \to \infty lim n^{4} (- 3 + \frac{9}{n ^{2}} + \frac{8}{n ^{4}}) = \infty \cdot (- 3 + 0 + 0) = \infty \cdot (- 3) = - \infty$

$f) n \to \infty lim (2 n^{5} - 7 n^{4} + n) = n \to \infty lim n^{5} (2 - \frac{7}{n} + \frac{1}{n ^{4}}) = \infty \cdot (2 - 0 + 0) = \infty \cdot 2 = \infty$