\(=\lim\limits_{x\rightarrow0}\left(x+2+\sqrt{3x+4}\right)\left[\frac{\frac{2x}{1+\sqrt{2x+1}}}{x^2+x}-\frac{sinx}{x^2+x}\right]\)
\(=\lim\limits_{x\rightarrow0}\left(x+2+\sqrt{3x+4}\right)\left[\frac{\frac{2}{1+\sqrt{2x+1}}}{x+1}-\frac{sinx}{x}.\frac{1}{\left(x+1\right)}\right]\)
\(=\left(2+\sqrt{4}\right)\left(\frac{1}{1}-\frac{1}{1}\right)=0\)
a. \(\lim\limits_{x\rightarrow0}\frac{\sqrt{1+2x}-1}{2x}\) f. \(\lim\limits_{x\rightarrow1}\frac{\sqrt{2x+7-3}}{2-\sqrt{x+3}}\)
b. \(\lim\limits_{x\rightarrow0}\frac{4x}{\sqrt{9+x}-3}\) g. \(\lim\limits_{x\rightarrow0}\frac{\sqrt{x^2+1}-1}{\sqrt{x^2+16}-4}\)
c. \(\lim\limits_{x\rightarrow2}\frac{\sqrt{x+7}-3}{x-2}\) h. \(\lim\limits_{x\rightarrow4}\frac{\sqrt{x+5}-\sqrt{2x+1}}{x-4}\)
d. \(\lim\limits_{x\rightarrow1}\frac{3x-2\sqrt{4x^2-x-2}}{x^2-3x+2}\) k. \(\lim\limits_{x\rightarrow0}\frac{\sqrt{x+1}+\sqrt{x+4}-3}{x}\)
e. \(\lim\limits_{x\rightarrow1}\frac{\sqrt{2x+7}+x-4}{x^3-4x^2+3}\)
Bài 1
a. \(\lim\limits_{x\rightarrow+\infty}\frac{1+2\sqrt{x}-x}{x+3}\) b. \(\lim\limits_{x\rightarrow+\infty}\frac{x^3+3x-1}{x^2\sqrt{x}+x}\) c. \(\lim\limits_{x\rightarrow-\infty}\frac{x+2\sqrt{1-x}}{1-x}\)
Bài 2: Tính các giới hạn sau biết \(\lim\limits_{x\rightarrow0}\frac{\sin x}{x}=1\)
a. \(\lim\limits_{x\rightarrow0}\frac{1-\cos x}{1-\cos3x}\) b. \(\lim\limits_{x\rightarrow0}\frac{\cot x-\sin x}{x^3}\) c. \(\lim\limits_{x\rightarrow\infty}\frac{x.\sin x}{2x^2}\)
\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt{4x+1}-\sqrt[3]{2x+1}}{x}\)
\(\lim\limits_{x\rightarrow1}\dfrac{\sqrt{4x+5}-3}{\sqrt[3]{5x+3}-2}\)
\(\lim\limits_{x\rightarrow-1}\dfrac{\sqrt[4]{2x+3}+\sqrt[3]{2+3x}}{\sqrt{x+2}-1}\)
tìm các giới hạn sau:
a, \(\lim\limits_{x\rightarrow-3}\frac{x+\sqrt{3-2x}}{x^2+3x}\)
b, \(\lim\limits_{x\rightarrow0}\frac{\sqrt{x+9}+\sqrt{x+16}-7}{x}\)
c, \(\lim\limits_{x\rightarrow\frac{1}{2}}\frac{8x^2-1}{6x^2-5x+1}\)
d, \(\lim\limits_{x\rightarrow0}\frac{\sqrt{x^2+1}-1}{4-\sqrt{x^2+16}}\)
tìm các giới hạn sau:
a; \(\lim\limits_{x\rightarrow\frac{\pi}{2}}\frac{sin\left(x-\frac{\pi}{4}\right)}{x}\)
b, \(\lim\limits_{x\rightarrow2}\frac{\sqrt[3]{3x^2-4}-\sqrt{3x-2}}{x+1}\)
c,\(\lim\limits_{x\rightarrow0}x^2sin\frac{1}{2}\)
\(\lim\limits_{x\rightarrow3}\dfrac{\sqrt{2x+3}-x}{x^2-4x+3}\)
\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt[3]{x+1}-1}{\sqrt[4]{2x+1}-1}\)
\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt{1+4x}-\sqrt[3]{1+6x}}{x^2}\)
\(\lim\limits_{x\rightarrow2}\dfrac{x-\sqrt{x+2}}{x-\sqrt[3]{3x+2}}\)
\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt{1+2x}-\sqrt[3]{1+3x}}{x^2}\)
\(\lim\limits_{x\rightarrow-1}\dfrac{\sqrt{5+4x}-\sqrt[3]{7+6x}}{x^3+x^2-x-1}\)
\(\lim\limits_{x\rightarrow-\infty}\left(\sqrt[3]{x^3+3x^2}-\sqrt{x^2-2x}\right)\)
\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt{1+2x}.\sqrt[3]{1+4x}-1}{x}\)
1) \(\lim\limits_{x\rightarrow0}\dfrac{2\sqrt{1+x}-\sqrt[3]{8-x}}{x}\)
2)\(\lim\limits_{x\rightarrow1}\dfrac{\sqrt[3]{x+7}-\sqrt{x+3}}{x^2-3x+2}\)
3)\(\lim\limits_{x\rightarrow1}\dfrac{\sqrt[3]{x^2+7}-\sqrt{5-x^2}}{x^2-1}\)
4)\(\lim\limits_{x\rightarrow-2}\dfrac{\sqrt{x+11}-\sqrt[3]{8x+43}}{2x^2+3x-2}\)
5) \(\lim\limits_{x\rightarrow0}\dfrac{\sqrt[n]{1+ax}-\sqrt[m]{1+bx}}{x}\)
6)\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt{1+4x}.\sqrt[3]{1+6x}-1}{x}\)
