Question

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Lim _{x->\(\infty\)} \(\left(1+\frac{1}{3x+1}\right)^{3x+1}\)

Lim _{x->\(\infty\)} \(\left(1+\frac{1}{x}\right)^{2x}\)

Lim _{x->\(\infty\)} \(\left(1+\frac{1}{X^2}\right)^{X^2}\)

Lim _{x->\(\infty\)}\(\left(1+\frac{1}{2x}\right)^X\)

Lim_{x->\(\infty\)}\(\left(1+\frac{1}{2x}\right)^{2x}\)

Answer 1

Đều thuộc dạng đưa về \(\lim\limits_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^x=e\)

\(\lim\limits_{x\rightarrow\infty}\left(1+\frac{1}{3x+1}\right)^{3x+1}=e\)

\(\lim\limits_{x\rightarrow\infty}\left(\left(1+\frac{1}{x}\right)^x\right)^2=\left(\lim\limits_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^x\right)^2=e^2\)

\(\lim\limits_{x\rightarrow\infty}\left(1+\frac{1}{x^2}\right)^{x^2}=e\)

\(\lim\limits_{x\rightarrow\infty}\left(1+\frac{1}{2x}\right)^x=\lim\limits_{x\rightarrow\infty}\left(1+\frac{1}{2x}\right)^{2x.\frac{1}{2}}=e^{\frac{1}{2}}=\sqrt{e}\)

\(\lim\limits_{x\rightarrow\infty}\left(1+\frac{1}{2x}\right)^{2x}=e\)