{"id":2130,"date":"2023-04-02T09:57:56","date_gmt":"2023-04-02T01:57:56","guid":{"rendered":"https:\/\/www.appblog.cn\/?p=2130"},"modified":"2023-04-05T20:41:27","modified_gmt":"2023-04-05T12:41:27","slug":"fundamentals-of-high-school-mathematics-limits-of-functions-and-the-origin-of-natural-constants-e","status":"publish","type":"post","link":"https:\/\/www.appblog.cn\/index.php\/2023\/04\/02\/fundamentals-of-high-school-mathematics-limits-of-functions-and-the-origin-of-natural-constants-e\/","title":{"rendered":"\u9ad8\u4e2d\u6570\u5b66\u57fa\u7840\uff1a\u51fd\u6570\u7684\u6781\u9650\u53ca\u81ea\u7136\u5e38\u6570e\u7684\u7531\u6765"},"content":{"rendered":"

\u51fd\u6570\u7684\u6781\u9650<\/h2>\n

1\u3001\u81ea\u53d8\u91cf\u8d8b\u4e8e\u6709\u9650\u5236$x_0$\u65f6\u51fd\u6570\u7684\u6781\u9650<\/p>\n

\uff081\uff09$x \\rightarrow x_0$   \uff082\uff09$x \\rightarrow x_0^+$  \uff083\uff09$x \\rightarrow x_0^-$<\/p>\n

<\/p>\n

2\u3001\u81ea\u53d8\u91cf\u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\u51fd\u6570\u7684\u6781\u9650<\/p>\n

\uff081\uff09$x \\rightarrow \u221e$  \uff082\uff09$x \\rightarrow +\u221e$  \uff083\uff09$x \\rightarrow -\u221e$<\/p>\n

\u81ea\u53d8\u91cf$x \\rightarrow x_0$\u65f6\u51fd\u6570\u6781\u9650\u7684\u5b9a\u4e49<\/h3>\n

\u8bbe\u51fd\u6570$f(x)$\u5728\u70b9$x_0$\u7684\u67d0\u4e00\u53bb\u5fc3\u9886\u57df\u5185\u6709\u5b9a\u4e49\uff0c\u5982\u679c\u5b58\u5728\u5e38\u6570$A$\uff0c\u5bf9\u4efb\u610f\u7ed9\u5b9a\u7684\u6b63\u6570$\\varepsilon$\uff08\u65e0\u8bba\u5b83\u6709\u591a\u5c11\uff09\uff0c\u603b\u5b58\u5728\u6b63\u6570$\\delta$\uff0c\u4f7f\u5f97\u5f53$x$\u6ee1\u8db3$0<|x-x_0|<\\delta$\u65f6\uff0c\u5bf9\u5e94\u7684\u51fd\u6570\u503c\u90fd\u6709$|f(x)-A|<\\varepsilon$\uff0c\u5219\u79f0$A$\u4e3a\u51fd\u6570$f(x)$\u5f53$x \\rightarrow x0$\u65f6\u7684\u6781\u9650\uff0c\u8bb0\u4f5c$\\lim \\limits<\/em>{x \\to x_0} f(x) = A$\u6216$f(x) \\rightarrow A \\, (x \\rightarrow x_0)$<\/p>\n

$\\varepsilon – \\delta$\u8bed\u8a00\u63cf\u8ff0\uff1a<\/p>\n

$\\lim \\limits_{x \\to x_0} f(x) = A \\Longleftrightarrow \\forall \\varepsilon > 0$\uff0c$\\exists \\delta > 0$\uff0c\u5f53$0 < |x-x_0| < \\delta$\u65f6\uff0c$|f(x)-A| < \\varepsilon$\u3002<\/p>\n

\u5de6\u6781\u9650\u4e0e\u53f3\u6781\u9650\uff08\u5355\u4fa7\u6781\u9650\uff09<\/h3>\n

\u5de6\u6781\u9650<\/h4>\n

$\\lim \\limits_{x \\to x_0^+} f(x) = A \\Longleftrightarrow \\forall \\varepsilon > 0$\uff0c$\\exists \\delta > 0$\uff0c\u5f53$x \\in (x_0-\\delta,x_0)$\u65f6\uff0c$|f(x)-A| < \\varepsilon$\u3002<\/p>\n

\u53f3\u6781\u9650<\/h4>\n

$\\lim \\limits_{x \\to x_0^-} f(x) = A \\Longleftrightarrow \\forall \\varepsilon > 0$\uff0c$\\exists \\delta > 0$\uff0c\u5f53$x \\in (x_0,x_0+\\delta)$\u65f6\uff0c$|f(x)-A| < \\varepsilon$\u3002<\/p>\n

\u5de6\u53f3\u6781\u9650\u76f8\u7b49<\/h4>\n

$\\lim \\limits_{x \\to x0} f(x) = A \\Longleftrightarrow \\lim \\limits<\/em>{x \\to x0^+} f(x) = A \\, \\lim \\limits<\/em>{x \\to x_0^-} f(x) = A$<\/p>\n

\u81ea\u53d8\u91cf$x \\rightarrow \u221e$\u65f6\u51fd\u6570\u6781\u9650\u7684\u5b9a\u4e49<\/h3>\n

\u8bbe\u51fd\u6570$f(x)$\u5728\u5f53$|x|$\u5927\u4e8e\u67d0\u4e00\u6b63\u6570\u65f6\u6709\u5b9a\u4e49\uff0c\u5982\u679c\u5b58\u5728\u5e38\u6570$A$\uff0c\u5bf9\u4efb\u610f\u7ed9\u5b9a\u7684\u6b63\u6570$\\varepsilon$\uff08\u65e0\u8bba\u5b83\u6709\u591a\u5c11\uff09\uff0c\u603b\u5b58\u5728\u6b63\u6570$X$\uff0c\u4f7f\u5f97\u5f53$x$\u6ee1\u8db3$|x|>X$\u65f6\uff0c\u5bf9\u5e94\u7684\u51fd\u6570\u503c\u90fd\u6709$|f(x)-A|<\\varepsilon$\uff0c\u5219\u79f0$A$\u4e3a\u51fd\u6570$f(x)$\u5f53$x \\rightarrow \u221e$\u65f6\u7684\u6781\u9650\uff0c\u8bb0\u4f5c$\\lim \\limits_{x \\to \u221e} f(x) = A$\u6216$f(x) \\rightarrow A \\, (x \\rightarrow \u221e)$<\/p>\n

$\\varepsilon – X$\u8bed\u8a00\u63cf\u8ff0\uff1a<\/p>\n

$\\lim \\limits_{x \\to \u221e} f(x) = A \\Longleftrightarrow \\forall \\varepsilon > 0$\uff0c$\\exists X > 0$\uff0c\u5f53$|x| > X$\u65f6\uff0c$|f(x)-A| < \\varepsilon$\u3002<\/p>\n

\u5de6\u6781\u9650\u4e0e\u53f3\u6781\u9650\uff08\u5355\u4fa7\u6781\u9650\uff09<\/h3>\n

$\\lim \\limits_{x \\to -\u221e} f(x) = A \\Longleftrightarrow \\forall \\varepsilon > 0$\uff0c$\\exists X > 0$\uff0c\u5f53$x < -X$\u65f6\uff0c$|f(x)-A| < \\varepsilon$\u3002<\/p>\n

$\\lim \\limits_{x \\to +\u221e} f(x) = A \\Longleftrightarrow \\forall \\varepsilon > 0$\uff0c$\\exists X > 0$\uff0c\u5f53$x > X$\u65f6\uff0c$|f(x)-A| < \\varepsilon$\u3002<\/p>\n

\u51fd\u6570\u6781\u9650\u7684\u6027\u8d28<\/h2>\n

\u51fd\u6570\u6781\u9650\u7684\u552f\u4e00\u6027<\/h3>\n

\u5b9a\u7406\uff1a\u5982\u679c$\\lim \\limits_{x \\to x_0} f(x)$\u5b58\u5728\uff0c\u5219\u6b64\u6781\u9650\u552f\u4e00\u3002<\/strong><\/p>\n

\u51fd\u6570\u6781\u9650\u7684\u5c40\u90e8\u6709\u754c\u6027<\/h3>\n

\u5b9a\u7406\uff1a\u82e5$\\lim \\limits_{x \\to x_0} f(x) = A$\uff0c\u5219\u5b58\u5728\u5e38\u6570$M > 0$\u548c$\\delta > 0$\uff0c\u4f7f\u5f97\u5f53$0 < |x-x_0| < \\delta$\u65f6\uff0c\u6709$|f(x)| \u2264 M$\u3002<\/strong><\/p>\n

\u8bc1\uff1a\u56e0$\\lim \\limits_{x \\to x_0} f(x) = A$\uff0c\u5219$\\forall \\varepsilon > 0$\uff0c$\\exists \\delta > 0$\uff0c\u4f7f\u5f97<\/p>\n

\u5f53$0 < |x-x_0| < \\delta$\u65f6\uff0c\u6709$|f(x) – A| \u2264 \\varepsilon$\u3002\u7279\u522b\u5730\u53d6$\\varepsilon = 1$\uff0c\u5219<\/p>\n

$|f(x)| = |f(x) – A + A| \u2264 1 + |A|$\u3002\u53d6$M = 1 + |A|$\u5373\u53ef\u3002<\/p>\n

\u51fd\u6570\u6781\u9650\u7684\u5c40\u90e8\u4fdd\u53f7\u6027<\/h3>\n

\u5b9a\u7406\uff1a\u5982\u679c$\\lim \\limits_{x \\to x_0} f(x) = A$\uff0c\u800c\u4e14$A>0 \\, (A<0)$\uff0c\u90a3\u4e48\u5b58\u5728\u5e38\u6570$\\delta > 0$\uff0c\u4f7f\u5f97\u5f53$0 < |x-x_0| < \\delta$\u65f6\uff0c$f(x)>0 \\, (f(x)<0)$\u3002<\/strong><\/p>\n

\u8bc1\uff1a\u53ea\u8bc1$A<0$\u7684\u60c5\u51b5\uff0c\u56e0\u4e3a$\\lim \\limits_{x \\to x_0} f(x) = A < 0$\uff0c\u53d6$\\varepsilon = -\\frac{A}{2}$\uff0c\u5219$\\exists \\delta \\gt 0$\uff0c\u5f53$0 < |x-x_0| < \\delta$\u65f6\uff0c\u6709<\/p>\n

$|f(x)-A| < -\\frac{A}{2}$\uff0c\u5373$f(x) < A-\\frac{A}{2}$\uff0c\u4ece\u800c<\/p>\n

$f(x) < \\frac{A}{2} < 0$<\/p>\n

\u4e24\u4e2a\u91cd\u8981\u6781\u9650<\/h2>\n

\u7b2c\u4e00\u4e2a\u91cd\u8981\u6781\u9650<\/h3>\n

$$
\n\\lim \\limits_{x \\to 0} \\frac{\\sin x}{x} = 1
\n$$<\/p>\n

\u8bc1\uff1a\u5f53$x \\in (0, \\frac{\\pi}{2})$\u65f6\uff0c\u6839\u636e\u5185\u542b\u4e09\u89d2\u5f62\u9762\u79ef < \u5706\u6247\u5f62\u9762\u79ef < \u5916\u5207\u4e09\u89d2\u5f62\u9762\u79ef<\/p>\n

$$
\n\\frac{1}{2}\u00b71\u00b71\u00b7\\sin x < frac{x}{2\\pi}\u00b7\\pi\u00b71^2 < \\frac{1}{2}\u00b71\u00b7\\tan x
\n$$<\/p>\n

$$
\n\u5373 \\sin x < x < \\tan x (0 < x < \\frac{\\pi}{2})
\n$$<\/p>\n

$$
\n\u4ece\u800c 1 < \\frac{x}{\\sin x} < \\frac{1}{\\cos x}$
\n$$<\/p>\n

$$
\n\u663e\u7136\u6709 \\cos x < \\frac{\\sin x}{x} < 1 (0 < |x| < \\frac{\\pi}{2})
\n$$<\/p>\n

$$
\n\u53c8 \\lim \\limits{x \\to 0} \\cos x = 1\uff0c\u6545\\lim \\limits<\/em>{x \\to 0} \\frac{\\sin x}{x} = 1
\n$$<\/p>\n

\u7b2c\u4e8c\u4e2a\u91cd\u8981\u6781\u9650<\/h3>\n

$$
\n\\lim \\limits_{x \\to \u221e} (1+\\frac{1}{x})^x = e
\n$$<\/p>\n

$$
\n\\lim \\limits_{x \\to 0} (1+x)^{1\/x} = e
\n$$<\/p>\n

\u8bc1\u660e\u601d\u8def\uff1a<\/p>\n

\uff081\uff09\u9996\u5148\u8bc1\u660e\u6570\u5217$\\{x_n\\}$\u662f\u5355\u8c03\u6709\u754c\u6570\u5217\u4ece\u800c\u6781\u9650\u5b58\u5728\uff0c\u5176\u4e2d
\n$$
\nxn = (1+\\frac{1}{n})^n
\n$$
\n\uff082\uff09\u5176\u6b21\u5229\u7528\u4e24\u8fb9\u5939\u51c6\u5219\u8bc1\u660e
\n$$
\n\\lim \\limits<\/em>{x \\to +\u221e} (1+\\frac{1}{x})^x = e
\n$$
\n\uff083\uff09\u518d\u7528\u53d8\u91cf\u4ee3\u6362\u6cd5\u8bc1\u660e
\n$$
\n\\lim \\limits_{x \\to -\u221e} (1+\\frac{1}{x})^x = e
\n$$
\n\uff084\uff09\u8054\u5408\u4e0a\u9762\u4e24\u4e2a\u7ed3\u8bba\u53ef\u5f97<\/p>\n

\u6570\u5217\u6536\u655b<\/h3>\n

\u5148\u8bc1\uff1a\u6570\u5217$\\{x_n\\}$\u6536\u655b\uff0c\u5176\u4e2d
\n$$
\nx_n = (1+\\frac{1}{n})^n
\n$$<\/p>\n

\u7b2c\u4e00\u6b65\uff0c\u8bc1\u660e\u6570\u5217$\\{x_n\\}$\u662f\u5355\u8c03\u589e\u52a0\u7684<\/p>\n

$$
\n\\begin{align}
\nx_n & = (1+\\frac{1}{n})^n \\
\n& = C_n^0(\\frac{1}{n})^0 + C_n^1(\\frac{1}{n})^1 + C_n^2(\\frac{1}{n})^2 + \u00b7\u00b7\u00b7 + C_n^n(\\frac{1}{n})^n \\
\n& = 1 + \\frac{n}{1!}\u00b7\\frac{1}{n} + \\frac{n(n-1)}{2!}\u00b7\\frac{1}{n^2} + \\frac{n(n-1)(n-2)}{3!}\u00b7\\frac{1}{n^3} + \u00b7\u00b7\u00b7 + \\frac{n(n-1)(n-2)\u00b7\u00b7\u00b72\u00b71}{n!}\u00b7\\frac{1}{n^n} \\
\n& = 1 + 1 + \\frac{1}{2!}(1-\\frac{1}{n}) + \\frac{1}{3!}(1-\\frac{1}{n})(1-\\frac{2}{n}) + \u00b7\u00b7\u00b7 + \\frac{1}{n!}(1-\\frac{1}{n})(1-\\frac{2}{n})\u00b7\u00b7\u00b7(1-\\frac{n-1}{n})
\n\\end{align}
\n$$<\/p>\n

\u540c\u7406<\/p>\n

$$
\n\\begin{align}
\nx_{n+1} &= 1 + 1 + \\frac{1}{2!}(1-\\frac{1}{n+1}) + \\frac{1}{3!}(1-\\frac{1}{n+1})(1-\\frac{2}{n+1}) + \u00b7\u00b7\u00b7 \\
\n&+ \\frac{1}{n!}(1-\\frac{1}{n+1})(1-\\frac{2}{n+1})\u00b7\u00b7\u00b7(1-\\frac{n-1}{n+1}) \\
\n&+ \\frac{1}{(n+1)!}(1-\\frac{1}{n+1})(1-\\frac{2}{n+1})\u00b7\u00b7\u00b7(1-\\frac{n}{n+1})
\n\\end{align}
\n$$<\/p>\n

\u6613\u89c1$x_{n+1} > x_n$\uff0c$\\forall n \\in N^+$<\/p>\n

\u7b2c\u4e00\u6b65\uff0c\u8bc1\u660e\u6570\u5217$\\{x_n\\}$\u6709\u754c<\/p>\n

$$
\nx_n=1 + 1 + \\frac{1}{2!}(1-\\frac{1}{n}) + \\frac{1}{3!}(1-\\frac{1}{n})(1-\\frac{2}{n}) + \u00b7\u00b7\u00b7 + \\frac{1}{n!}(1-\\frac{1}{n})(1-\\frac{2}{n})\u00b7\u00b7\u00b7(1-\\frac{n-1}{n})
\n$$<\/p>\n

\u4ece\u800c<\/p>\n

$$
\n\\begin{align}
\nx_n &< 1 + 1 + \\frac{1}{2!} + \\frac{1}{3!} + \u00b7\u00b7\u00b7 + \\frac{1}{n!} \\
\n&< 1 + 1 + \\frac{1}{2} + \\frac{1}{2^2} + \u00b7\u00b7\u00b7 + \\frac{1}{2^(n-1)} \\
\n&= 1 + \\frac{1\u00b7(1-(1-\\frac{1}{2})^n)}{1-\\frac{1}{2}} \\
\n&= 1 + 2(1-(\\frac{1}{2})^n) \\
\n&< 3
\n\\end{align}
\n$$<\/p>\n

\u5176\u4e2d\u6211\u4eec\u7528\u4e86\u4e0d\u7b49\u5f0f$2^{n-1}\u2264n!$ (\u6570\u5b66\u5f52\u7eb3\u6cd5)<\/p>\n

\u4e8e\u662f\uff0c\u7531\u5355\u8c03\u589e\u52a0\u548c\u6709\u754c\u6027\u77e5\u6570\u5217$\\{x_n\\}$\u6781\u9650\u5b58\u5728\uff0c\u8bb0<\/p>\n

$$
\n\\lim \\limits_{n \\to \u221e} (1+\\frac{1}{n})^n = e
\n$$<\/p>\n

\u4e0b\u8bc1\uff1a\u51fd\u6570\u6781\u9650<\/p>\n

$$
\n\\lim \\limits_{x \\to \u221e} (1+\\frac{1}{x})^x = e
\n$$<\/p>\n

\u4e00\u65b9\u9762\uff0c\u5f53$x>1$\u65f6\uff0c\u8bbe$n\u2264x<n+1$\uff0c\u5219<\/p>\n

$$
\n(1+\\frac{1}{n+1})^n < (1+\\frac{1}{x+1})^x < (1+\\frac{1}{n})^{n+1}
\n$$<\/p>\n

\u7531\u4e8e<\/p>\n

$$
\n\\lim \\limits{x \\to \u221e} (1+\\frac{1}{n+1})^n = \\lim \\limits<\/em>{x \\to \u221e} \\frac{(1+\\frac{1}{n+1})^{n+1}}{1+\\frac{1}{n+1}} = e
\n$$<\/p>\n

$$
\n\\lim \\limits{x \\to \u221e} (1+\\frac{1}{n})^{n+1} = \\lim \\limits<\/em>{x \\to \u221e} (1+\\frac{1}{n})^n(1+\\frac{1}{n}) = e
\n$$<\/p>\n

\u56e0$n \\rightarrow +\u221e$\u65f6\uff0c$x \\rightarrow +\u221e$\uff0c\u6545<\/p>\n

$$
\n\\lim \\limits_{x \\to +\u221e} (1+\\frac{1}{x})^x = e
\n$$<\/p>\n

\u53e6\u4e00\u65b9\u9762\uff0c\u5f53$x \\rightarrow -\u221e$\u65f6\uff0c\u4ee4$x=-(t+1)$\uff0c\u5219$t \\rightarrow +\u221e$\u65f6\uff0c\u4ece\u800c\u6709<\/p>\n

\\begin{align}
\n\\lim \\limits{x \\to -\u221e} (1+\\frac{1}{x})^x & = \\lim \\limits<\/em>{x \\to +\u221e} (1-\\frac{1}{t+1})^{-(t+1)} \\
\n& = \\lim \\limits{x \\to +\u221e} (\\frac{t}{t+1})^{-(t+1)} \\
\n& = \\lim \\limits<\/em>{x \\to +\u221e} (1+\\frac{1}{t})^{t+1} \\
\n& = \\lim \\limits_{x \\to +\u221e} (1+\\frac{1}{t})^t(1+\\frac{1}{t}) \\
\n& = e
\n\\end{align}<\/p>\n

\u56e0<\/p>\n

$$
\n\\lim \\limits{x \\to +\u221e} (1+\\frac{1}{x})^x = \\lim \\limits<\/em>{x \\to -\u221e} (1+\\frac{1}{x})^x = e
\n$$<\/p>\n

\u6545<\/p>\n

$$
\n\\lim \\limits_{x \\to \u221e} (1+\\frac{1}{x})^x = e
\n$$<\/p>\n

\u81ea\u7136\u5e38\u6570e<\/h3>\n

\uff081\uff09\u5047\u8bbe\u67d0\u79cd\u7c7b\u7684\u5355\u7ec6\u80de\u751f\u7269\u6bcf24\u5c0f\u65f6\u5206\u88c2\u4e00\u6b21\uff0c\u5728\u4e0d\u8003\u8651\u6b7b\u4ea1\u4e0e\u53d8\u5f02\u7684\u60c5\u51b5\u4e0b\uff0c\u90a3\u4e48\u5f88\u663e\u7136\u8fd9\u7fa4\u5355\u7ec6\u80de\u751f\u7269\u7684\u603b\u6570\u91cf\u6bcf\u5929\u4f1a\u589e\u52a0\u4e00\u500d\uff0c\u4e5f\u5c31\u662f\u589e\u957f\u7387\u4e3a<\/p>\n

$$
\ngrouth = (1+100%) = 2
\n$$<\/p>\n

\uff082\uff09\u5047\u8bbe\u8fd9\u79cd\u7ec6\u80de\u6bcf\u8fc712\u5c0f\u65f6\uff0c\u5e73\u5747\u4f1a\u4ea7\u751f\u4e00\u534a\u7684\u539f\u6570\u91cf\u7684\u65b0\u7ec6\u80de\uff0c\u800c\u4e14\u65b0\u7ec6\u80de\u5728\u4e4b\u540e\u768412\u5c0f\u65f6\u4e5f\u4f1a\u5b58\u5728\u5206\u88c2\uff0c\u90a3\u4e48\u589e\u957f\u7387\u4e3a<\/p>\n

$$
\ngrouth = (1+\\frac{100%}{2})^2 = 2.25
\n$$<\/p>\n

\uff083\uff09\u5047\u8bbe\u6bcf\u8fc78\u5c0f\u65f6\u5206\u88c2\u4e00\u6b21\uff0c\u90a3\u4e48\u589e\u957f\u7387\u4e3a<\/p>\n

$$
\ngrouth = (1+\\frac{100%}{3})^3 = 2.37037…
\n$$<\/p>\n

\uff084\uff09\u5b9e\u9645\u4e0a\uff0c\u7ec6\u80de\u7684\u5206\u88c2\u662f\u4e0d\u95f4\u65ad\u7684\u3001\u8fde\u7eed\u7684\uff0c\u65f6\u523b\u90fd\u4f1a\u4ea7\u751f\u65b0\u7ec6\u80de\uff0c\u800c\u4e14\u6bcf\u4e2a\u65b0\u7ec6\u80de\u90fd\u4f1a\u7acb\u5373\u548c\u6bcd\u4f53\u7ec6\u80de\u4e00\u8d77\u8fdb\u884c\u5206\u88c2\u64cd\u4f5c\uff0c\u90a3\u4e48\u4e00\u592924\u5c0f\u65f6\u5206\u88c2\u7684\u589e\u957f\u7387\u4e3a<\/p>\n

$$
\ngrouth = (1+\\frac{100%}{n})^n = 2.71828…
\n$$<\/p>\n

\uff085\uff09\u56e0\u6b64\u5c06\u8be5\u503c\u79f0\u4e3a$e$\uff0c\u8868\u793a\u5355\u4f4d\u65f6\u95f4\u5185\uff0c\u6301\u7eed\u7ffb\u500d\u589e\u957f\u6240\u80fd\u8fbe\u5230\u7684\u6781\u9650\u503c\uff0c\u516c\u5f0f\u4e3a<\/p>\n

$$
\ne = \\lim \\limits_{n \\to \u221e} (1+\\frac{1}{n})^n
\n$$<\/p>\n

\u5e38\u89c1\u8ba1\u7b97<\/h3>\n

$$
\n\\lim \\limits{x \\to 0} \\frac{\\sin 2x}{x} = \\lim \\limits<\/em>{x \\to 0} \\frac{2\\sin 2x}{2x} = 2
\n$$<\/p>\n

$$
\n\\lim \\limits{x \\to 0} \\frac{\\tan x}{x} = \\lim \\limits<\/em>{x \\to 0} \\frac{\\sin x}{x} \\frac{1}{\\cos x} = 1
\n$$<\/p>\n

$$
\n\\lim \\limits{x \\to \u221e} (1+\\frac{1}{2x})^x = \\lim \\limits<\/em>{x \\to \u221e} ((1+\\frac{1}{2x})^{2x})^{\\frac{1}{2}} = e^{\\frac{1}{2}} = \\sqrt{e}
\n$$<\/p>\n

$$
\n\\lim \\limits{x \\to \u221e} (1-\\frac{1}{x})^x = \\lim \\limits<\/em>{x \\to \u221e} ((1+\\frac{1}{-x})^{-x})^{-1} = e^{-1} = \\frac{1}{e}
\n$$<\/p>\n","protected":false},"excerpt":{"rendered":"

\u51fd\u6570\u7684\u6781\u9650 1\u3001\u81ea\u53d8\u91cf\u8d8b\u4e8e\u6709\u9650\u5236$x_0$\u65f6\u51fd\u6570\u7684\u6781\u9650 \uff081\uff09$x \\rightarrow x_0$ &nbs […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[509],"tags":[],"class_list":["post-2130","post","type-post","status-publish","format-standard","hentry","category-mathematics-fundamentals"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2130","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2130"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2130\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2130"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2130"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}