Bài 1
a. \(\lim\limits_{x\rightarrow-1}\frac{x^5+1}{x^3+1}\)
b. \(\lim\limits_{x\rightarrow1}\frac{x^6-5x^5+x}{\left(1-x\right)^2}\)
c. \(\lim\limits_{x\rightarrow0}\frac{\left(1+x\right)\left(1+2x\right)\left(1+3x\right)-1}{x}\)
d. \(\lim\limits_{x\rightarrow0}\frac{\left(1+x\right)^5-\left(1+x\right)}{x^5+x^2}\)
Bài 2
a. \(\lim\limits_{x\rightarrow1}\frac{x^m-1}{x^n-1}\)
b. \(\lim\limits_{x\rightarrow a}\frac{x-a}{x^n-a^n}\left(n\in Z^+,a\ne0\right)\)
c....
Đọc tiếp
Bài 1
a. \(\lim\limits_{x\rightarrow-1}\frac{x^5+1}{x^3+1}\)
b. \(\lim\limits_{x\rightarrow1}\frac{x^6-5x^5+x}{\left(1-x\right)^2}\)
c. \(\lim\limits_{x\rightarrow0}\frac{\left(1+x\right)\left(1+2x\right)\left(1+3x\right)-1}{x}\)
d. \(\lim\limits_{x\rightarrow0}\frac{\left(1+x\right)^5-\left(1+x\right)}{x^5+x^2}\)
Bài 2
a. \(\lim\limits_{x\rightarrow1}\frac{x^m-1}{x^n-1}\)
b. \(\lim\limits_{x\rightarrow a}\frac{x-a}{x^n-a^n}\left(n\in Z^+,a\ne0\right)\)
c. \(\lim\limits_{x\rightarrow0}\frac{x+x^2+...+x^n-n}{x-1}\)
d. \(\lim\limits_{x\rightarrow0}\frac{\left(1+x\right)\left(1+2x\right)...\left(1+nx\right)-1}{x}\)
Bài 1:
\(a=\lim\limits_{x\rightarrow-1}\frac{\left(x+1\right)\left(x^4-x^3+x^2-x+1\right)}{\left(x+1\right)\left(x^2-x+1\right)}=\lim\limits_{x\rightarrow-1}\frac{x^4-x^3+x^2-x+1}{x^2-x+1}=\frac{5}{3}\)
\(b=\frac{1-5+1}{0}=\frac{-3}{0}=-\infty\)
\(c=\lim\limits_{x\rightarrow1}\frac{x\left(1+2x\right)\left(1+3x\right)+2x\left(1+3x\right)+3x}{x}=\lim\limits_{x\rightarrow1}\left[\left(1+2x\right)\left(1+3x\right)+2\left(1+3x\right)+3\right]=1+2+3=6\)
\(d=\lim\limits_{x\rightarrow0}\frac{5\left(1+x\right)^4-1}{5x^4+2x}=\frac{4}{0}=+\infty\)
Bài 2:
\(a=\lim\limits_{x\rightarrow1}\frac{x^m-1}{x^n-1}=\lim\limits_{x\rightarrow1}\frac{mx^{m-1}}{nx^{n-1}}=\frac{m}{n}\)
\(b=\lim\limits_{x\rightarrow a}\frac{x-a}{x^n-a^n}=\lim\limits_{x\rightarrow a}\frac{1}{nx^{n-1}}=\frac{1}{n.a^{n-1}}\)
\(c=\lim\limits_{x\rightarrow0}\frac{x+x^2+...+x^n-n}{x-1}=\frac{-n}{-1}=n\)
\(\left(1+x\right)\left(1+2x\right)...\left(1+nx\right)=x\left(1+2x\right)...\left(1+nx\right)+\left(1+2x\right)\left(1+3x\right)...\left(1+nx\right)\)
\(=x\left(1+2x\right)...\left(1+nx\right)+2x\left(1+3x\right)...\left(1+nx\right)+\left(1+3x\right)...\left(1+nx\right)\)
\(=...\)
\(=x\left(1+2x\right)...\left(1+nx\right)+2x\left(1+3x\right)...\left(1+nx\right)+...+nx+1\)
\(\Rightarrow\lim\limits_{x\rightarrow0}\frac{\left(1+2x\right)\left(1+3x\right)...\left(1+nx\right)-1}{x}\)
\(=\lim\limits_{x\rightarrow0}\frac{x\left(1+2x\right)...\left(1+nx\right)+2x\left(1+3x\right)...\left(1+nx\right)+...+nx}{x}\)
\(=\lim\limits_{x\rightarrow0}\left[\left(1+2x\right)...\left(1+nx\right)+2\left(1+3x\right)...\left(1+nx\right)+...+n\right]\)
\(=1+2+3+...+n=\frac{n\left(n+1\right)}{2}\)