Question

[Huỳnh Thị Ngọc Nhi]

1) lim(2x-1-\(\sqrt{4x^2-4x-3}\))

2) lim\(\dfrac{\sqrt{2x^2-2}-\sqrt{4x-3}+2x-7}{9-x^2}\)

3) lim(\(x^3-1\sqrt{\dfrac{x}{x^2-1}}\)

Giúp giùm mình đi mấy bạn

Answer 1

[Akai Haruma]

Câu 1:

\(\lim _{x\to +\infty}(2x-1-\sqrt{4x^2-4x-3})=\lim_{x\to +\infty}\frac{(2x-1)^2-(4x^2-4x-3)}{2x-1+\sqrt{4x^2-4x-3}}\) (liên hợp)

\(=\lim_{x\to +\infty}\frac{4}{2x-1+\sqrt{4x^2-4x-3}}=4\lim_{x\to +\infty}\frac{1}{2x-1+\sqrt{4x^2-4x-3}}\)

Ta thấy với \(x\to +\infty\Rightarrow 2x-1+\sqrt{4x^2-4x-3}\to +\infty\)

Do đó: \(\lim_{x\to +\infty}\frac{1}{2x-1+\sqrt{4x^2-4x-3}}=0\) (theo dạng \(\lim _{t\to \infty}\frac{1}{t}=0\) )

\(\Rightarrow \lim _{x\to +\infty}(2x-1-\sqrt{4x^2-4x-3})=0\)

Câu 3:

\(\lim_{x\to 1+} (x^3-1)\sqrt{\frac{x}{x^2-1}}=\lim_{x\to 1+}(x^2+x+1)\sqrt{\frac{x(x-1)^2}{x^2-1}}\)

\(=\lim_{x\to 1+}(x^2+x+1)\sqrt{\frac{x(x-1)}{x+1}}=(1+1+1)\sqrt{\frac{1.0}{1+1}}=0\)

 

Answer 2

[Akai Haruma]

Câu 2:

\(\lim_{x\to 3}\frac{\sqrt{2x^2-2}-\sqrt{4x-3}+2x-7}{9-x^2}=\lim_{x\to 3}\frac{\sqrt{2x^2-2}-4}{9-x^2}-\lim_{x\to 3}\frac{\sqrt{4x-3}-3}{9-x^2}+\lim_{x\to 3}\frac{2x-6}{9-x^2}\)

Ta có:

\(\lim_{x\to 3}\frac{2x^2-2-16}{(\sqrt{2x^2-2}+4)(9-x^2)}=\lim_{x\to 3}\frac{2(x^2-9)}{(\sqrt{2x^2-2}+4)(9-x^2)}=\lim_{x\to 3}\frac{-2}{\sqrt{2x^2-2}+4}=\frac{-1}{4}\) (1)

\(\lim_{x\to 3}\frac{\sqrt{4x-3}-3}{9-x^2}=\lim_{x\to 3}\frac{4x-3-9}{(\sqrt{4x-3}+3)(9-x^2)}=\lim_{x\to 3}\frac{4(x-3)}{(\sqrt{4x-3}+3)(9-x^2)}\)

\(=\lim_{x\to 3}\frac{-4}{(\sqrt{4x-3}+3)(3+x)}=-\frac{1}{9}\) (2)

\(\lim _{x\to 3}\frac{2x-6}{9-x^2}=\lim_{x\to 3}\frac{2(x-3)}{9-x^2}=\lim_{x\to 3}\frac{-2}{x+3}=\frac{-1}{3}\) (3)

Từ \((1); (2); (3)\Rightarrow \lim_{x\to 3}\frac{\sqrt{2x^2-2}-\sqrt{4x-3}+2x-7}{9-x^2}=\frac{-1}{4}+\frac{1}{9}-\frac{1}{3}=\frac{-17}{36}\)