Question

[Nguyễn Kiều Anh]

Câu 1:

Giới hạn lim\(\dfrac{5\sqrt{3n^2+n}}{2\left(3n+2\right)}=\dfrac{a\sqrt{3}}{b}\)(a/b) khi đó tổng a+b bằng?

Câu 2:

Cho a và b là các số thực khác 0. Nếu lim_x->2 \(\dfrac{x^2+ax+b}{x-2}=6\) thì a+b bầng?

Answer 1

[Nguyễn Việt Lâm]

1.

\(\lim\dfrac{5\sqrt{3n^2+n}}{2\left(3n+2\right)}=\lim\dfrac{5\sqrt{3+\dfrac{1}{n}}}{2\left(3+\dfrac{2}{n}\right)}=\dfrac{5\sqrt{3}}{6}\Rightarrow a+b=11\)

2.

\(\lim\limits_{x\rightarrow2}\dfrac{x^2+ax+b}{x-2}=6\) khi \(x^2+ax+b=0\) có nghiệm \(x=2\)

\(\Rightarrow4+2a+b=0\Rightarrow b=-2a-4\)

\(\lim\limits_{x\rightarrow2}\dfrac{x^2+ax-2a-4}{x-2}=\lim\limits_{x\rightarrow2}\dfrac{\left(x-2\right)\left(x+2\right)+a\left(x-2\right)}{x-2}=\lim\limits_{x\rightarrow2}\dfrac{\left(x-2\right)\left(x+a+2\right)}{x-2}\)

\(=\lim\limits_{x\rightarrow2}\left(x+a+2\right)=a+4\Rightarrow a+4=6\Rightarrow a=2\Rightarrow b=-8\)

\(\Rightarrow a+b=-6\)