Tính \(\lim\limits_{x\rightarrow0}\left(1+\sin\pi x\right)^{\cot\pi x}\). 1 e \(\frac{1}{e}\) \(e^{\pi}\) Hướng dẫn giải: \(\lim\limits_{x\rightarrow0}\left(1+\sin\pi x\right)^{\cot\pi x}=\lim\limits_{x\rightarrow0}\left[\left(1+\sin\pi x\right)^{\dfrac{1}{\sin\pi x}}\right]^{\cos\pi x}=\left(e\right)^{-1}=\dfrac{1}{e}\).
Tính \(\lim\limits_{x\rightarrow0}\left(1+\sin x\right)^{\frac{1}{x}}\). e \(\frac{1}{e}\) 1 \(\sqrt{e}\) Hướng dẫn giải: \(\lim\limits_{x\rightarrow0}\left(1+\sin x\right)^{\frac{1}{x}}=\lim\limits_{x\rightarrow0}\left[\left(1-2\sin x\right)^{\frac{1}{\sin x}}\right]^{-\frac{\sin x}{x}}=e^1=e\)
Tính \(\lim\limits_{x\rightarrow0}\left(\cos x\right)^{\frac{1}{x^2}}\). e \(e^2\) \(\sqrt{e}\) \(\frac{1}{e}\) Hướng dẫn giải:
Tính \(\lim\limits_{x\rightarrow0}\frac{\sqrt{1+2x}-1}{2x}\) . 1 2 \(\frac{1}{2}\) 0 Hướng dẫn giải: \(\lim\limits_{x\rightarrow0}\frac{\sqrt{1+2x}-1}{2x}=\lim\limits_{x\rightarrow0}\frac{2x}{2x\left(\sqrt{1+2x}+1\right)}=\lim\limits_{x\rightarrow0}\frac{1}{\sqrt{1+2x}+1}=\frac{1}{2}\)
Tìm \(\lim\limits_{x\rightarrow0}\frac{4x}{\sqrt{x+9}-3}\) . 6 12 18 24 Hướng dẫn giải: \(\lim\limits_{x\rightarrow0}\frac{4x}{\sqrt{x+9}-3}=\lim\limits_{x\rightarrow0}\frac{4x\left(\sqrt{x+9}+3\right)}{x}\)\(=\lim\limits_{x\rightarrow0}\frac{4\left(\sqrt{x+9}+3\right)}{1}=24\)
Giá trị \(\lim\limits_{x\rightarrow7}\frac{2-\sqrt{x-3}}{x^2-49}\) bằng bao nhiêu? \(-\frac{1}{14}\) \(-\frac{1}{28}\) \(-\frac{1}{56}\) \(-\frac{1}{63}\) Hướng dẫn giải:
Tính \(\lim\limits_{x\rightarrow4}\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}}\) . -3 \(-\frac{1}{3}\) 3 \(\frac{1}{3}\) Hướng dẫn giải:
Tính \(\lim\limits_{x\rightarrow1}\frac{\sqrt{2x+7}+x-4}{x^3-4x^2+3}\) . \(\frac{4}{15}\) \(\frac{2}{5}\) \(-\frac{4}{15}\) \(-\frac{2}{5}\) Hướng dẫn giải:
Tính \(\lim\limits_{x\rightarrow\infty}\left(\frac{3x-2}{3x+1}\right)^{2x}\). \(e\) \(\frac{1}{e}\) \(e^2\) \(\frac{1}{e^2}\) Hướng dẫn giải: \(\lim\limits_{x\rightarrow\infty}\left(\frac{3x-2}{3x+1}\right)^{2x}=\lim\limits_{x\rightarrow\infty}\left[\left(1+\frac{-3}{3x+1}\right)^{\frac{3x+1}{-3}}\right]^{\frac{-3.2x}{3x+1}}=e^{-2}=\frac{1}{e^2}\)
Giá trị của \(\lim\limits_{x\rightarrow\infty}\left(\frac{x^2+1}{x^2-2}\right)^{x^2}\) bằng bao nhiêu? \(e^{-3}\) \(e^3\) \(e^{-2}\) \(e^2\) Hướng dẫn giải: \(\lim\limits_{x\rightarrow\infty}\left(\frac{x^2+1}{x^2-2}\right)^{x^2}=\lim\limits_{x\rightarrow\infty}\left[\left(1+\frac{3}{x^2-2}\right)^{\frac{x^2-2}{3}}\right]^{\frac{3x^2}{x^2-2}}=e^3\)