两个重要极限及相关推导极限 |
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两个重要极限: ① lim x → 0 sin x x = 1 \lim_{x \to 0}\frac{\sin x}{x} = 1 x→0limxsinx=1 ② lim x → ∞ ( 1 + 1 x ) x = e \lim_{x \to \infty}(1 + \frac{1}{x})^x = e x→∞lim(1+x1)x=e 关于重要极限①的推导极限可以参考: 无穷小的等价代换 关于重要极限②的由来可以参考:自然常数e与重要极限 由重要极限②可以推导出: lim x → ∞ ( 1 + 1 x ) x ⇒ lim x → 0 ( 1 + x ) 1 x = e \lim_{x \to \infty}(1 + \frac{1}{x})^x \Rightarrow \lim_{x \to 0}(1 + x)^{\frac{1}{x}} = e x→∞lim(1+x1)x⇒x→0lim(1+x)x1=e 重要极限②的例题 例:求 lim x → ∞ ( 1 + 2 x + 3 ) x \lim_{x \to \infty} (1 + \frac{2}{x + 3})^x limx→∞(1+x+32)x 解: lim x → ∞ ( 1 + 2 x + 3 ) x \lim_{x \to \infty} (1 + \frac{2}{x + 3})^x x→∞lim(1+x+32)x = lim x → ∞ ( 1 + 1 x + 3 2 ) ( x + 3 2 ) ⋅ 2 − 3 = \lim_{x \to \infty} (1 + \frac{1}{\frac{x + 3}{2}})^{(\frac{x+3}{2})·2 - 3} =x→∞lim(1+2x+31)(2x+3)⋅2−3 = lim x → ∞ ( 1 + 1 x + 3 2 ) x + 3 2 ⋅ 2 ⋅ ( 1 + 1 x + 3 2 ) − 3 = \lim_{x \to \infty} (1 + \frac{1}{\frac{x + 3}{2}})^{\frac{x+3}{2}·2} ·(1 + \frac{1}{\frac{x + 3}{2}})^{-3} =x→∞lim(1+2x+31)2x+3⋅2⋅(1+2x+31)−3 = lim x → ∞ ( 1 + 1 x + 3 2 ) x + 3 2 ⋅ 2 =\lim_{x \to \infty} (1 + \frac{1}{\frac{x + 3}{2}})^{\frac{x+3}{2}·2} =x→∞lim(1+2x+31)2x+3⋅2 = e 2 = e^2 =e2 关于重要极限②,还有一个变体例题,很容易迷惑人,但实际上并不是用重要极限的方法来做 例:求 lim x → ∞ ( 1 + 2 x ) 1 x \lim_{x \to \infty}(1+2x)^{\frac{1}{x}} x→∞lim(1+2x)x1 解: ( 1 + 2 x ) 1 x = e ln ( 1 + 2 x ) 1 x = e ln ( 1 + 2 x ) x (1+2x)^{\frac{1}{x}} = e^{\ln(1+2x)^{\frac{1}{x}}} = e^{\frac{\ln (1+2x)}{x}} (1+2x)x1=eln(1+2x)x1=exln(1+2x) 所以: lim x → ∞ ( 1 + 2 x ) 1 x = lim x → ∞ e ln ( 1 + 2 x ) x = e lim x → ∞ ln ( 1 + 2 x ) x \lim_{x \to \infty}(1+2x)^{\frac{1}{x}} = \lim_{x \to \infty}e^{\frac{\ln (1+2x)}{x}} = e^{\lim_{x \to \infty} \frac{\ln (1+2x)}{x}} x→∞lim(1+2x)x1=x→∞limexln(1+2x)=elimx→∞xln(1+2x) 应用洛必达法则,得: e lim x → ∞ ln ( 1 + 2 x ) x = e lim x → ∞ 2 1 + 2 x = e 0 = 1 e^{\lim_{x \to \infty} \frac{\ln (1+2x)}{x}} = e^{\lim_{x \to \infty} \frac{2}{1+2x}} = e^0 = 1 elimx→∞xln(1+2x)=elimx→∞1+2x2=e0=1 |
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