Question

Afla a, b, c știind limita

Answer 1

$l=\lim\limits_{n\to\infty} n(an+\sqrt{cn^2+bn+2}) = \lim\limits_{n\to\infty} n^2\Big(an+n\sqrt{c+\frac{b}{n}+\frac{2}{n}}\Big)=\\ \\ = \lim\limits_{n\to\infty} n(a+\sqrt{c+\frac{b}{n}+\frac{2}{n}})\\ \\ \dfrac{1}{n} = t \Rightarrow t\to 0\\ \\ l =\lim\limits_{t\to 0}\dfrac{a+\sqrt{c+bt+2t^2}}{t^2}\\ \\ \Rightarrow a = -\sqrt{c}\quad (\text{pentru a exista nedeterminarea 0 pe 0.)}\\ \\ l = \dfrac{0+\dfrac{4t+b}{2\sqrt{c+bt+2t^2}}}{2t} = \dfrac{4t+b}{4t\sqrt{c+bt+2t^2}}$

$\Rightarrow b = 0\quad (\text{pentru a exista nedeterminarea 0 pe 0.})\\ \\ l = \dfrac{4t}{4t\sqrt{c+bt+2t^2}} = \dfrac{4}{4\sqrt{c+bt+2t^2}+4t\cdot \dfrac{4t+b}{2\sqrt{c+bt+2t^2}}}\\ \\ \Rightarrow \dfrac{4}{4\sqrt{c}} = 1 \Rightarrow c = 1\\ \\\Rightarrow a = -\sqrt{1} = -1\\ \\\\ \Rightarrow a = -1,\quad b = 0,\quad c= 1$