\(lim\dfrac{\sqrt{n^3+1}-1}{n^2+1}\\ =lim\dfrac{\sqrt{\dfrac{n^3}{n^3}+\dfrac{1}{n^3}}-\dfrac{1}{n^3}}{n^2\left(1+\dfrac{1}{n^2}\right)}\\ =lim\dfrac{n\sqrt{n}.\left(\sqrt{1+\dfrac{1}{n^3}}-\dfrac{1}{n^3}\right)}{n\sqrt{n^2}\left(1+\dfrac{1}{n^2}\right)}\\ =lim\dfrac{1}{\sqrt{n}}.\dfrac{\sqrt{1+\dfrac{1}{n^3}}-\dfrac{1}{n^3}}{1+\dfrac{1}{n^2}}\\ =0\)
Tính các giới hạn sau:\(M=\lim\limits_{x\rightarrow0}\dfrac{\sqrt{1+4x}-\sqrt[3]{1+6x}}{1-cos3x}\)
\(N=\lim\limits_{X\rightarrow0}\dfrac{\sqrt[m]{1+ax}-\sqrt[n]{1+bx}}{\sqrt{1+x}-1}\)
\(V=\lim\limits_{x\rightarrow0}\dfrac{\left(1+mx\right)^n-\left(1+nx\right)^m}{\sqrt{1+2x}-\sqrt[3]{1+3x}}\)
Chứng minh các giới hạn sau :
\(1.lim\dfrac{n^2+1}{n}=+\infty\)
\(2.lim\dfrac{2-n}{\sqrt{n}}=-\infty\)
Tính các giới hạn sau:\(I_1=\lim\limits_{x\rightarrow1}\dfrac{\left(1-\sqrt{x}\right)\left(1-\sqrt[3]{x}\right)....\left(1-\sqrt[n]{x}\right)}{\left(1-x\right)^{n-1}}\)
\(I_2=\lim\limits_{x\rightarrow0}\dfrac{\left(\sqrt{1+x^2}+x\right)^n-\left(\sqrt{1+x^2}-x\right)^n}{x}\)
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)\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt{1+4x}.\sqrt[3]{1+6x}-1}{x}\)
Giúp em bài này với ạ!
Tìm các giới hạn sau:
a) \(\lim\limits_{x\to 1+} \dfrac{\sqrt{x}+\sqrt{x-1}-1}{\sqrt{x^2-1}}\)
b) \(\lim\limits_{x\to 1} \dfrac{x^n-1}{x^m-1}\) (m,n nguyên dương)
c) \(\lim\limits_{x\to 1} \dfrac{x+x^2+...+x^n-n}{x-1}\) (n nguyên dương)
Em cảm ơn rất nhiều!
tính giới hạn sau:
\(lim\dfrac{\sqrt{4n^2-n}+\sqrt[3]{8n^3+n^2}}{2n+3}\)
tìm các giới hạn sau:
a, lim\(\frac{2^{5n+1}+3}{3^{5n+2}+1}\)
b, lim\(\frac{\left(-1\right)^n+4.3^n}{\left(-1\right)^{n+1}-2.3^n}\)
c, lim \(\left(1+n^2-\sqrt{n^4+n}\right)\)
d, lim \(\frac{2cosn^2}{n^2+1}\)
e, lim \(\left(\sqrt{n^2-2}-\sqrt[3]{n^3+2n}\right)\)
Tính
1, \(lim\sqrt{n}\left(\sqrt{n+1}-\sqrt{n}\right)\)
2, \(lim\left(\sqrt[3]{n^3+1}-n\right)\)
tìm các giới hạn sau:
a; lim\(\frac{1+2+3+...+n}{3n^3}\)
b, lim \(\left(\frac{n+2}{n+1}+\frac{sin\text{n}}{2^n}\right)\)
c, lim \(\left(\sqrt{n^2-3n}-\sqrt{n^2+1}\right)\)
d,\(lim\left(\sqrt[3]{n^3+3n^2}-n\right)\)
