lim 极限计算器Limit Calculator
求函数的极限 lim f(x)：支持趋近于某个数、分数、pi 或 ±∞，可选双侧、左极限、右极限；符号计算给出精确值并用数值核对，能识别左右极限不相等、趋于无穷、振荡和无定义等“极限不存在”的情况。Find the limit of a function, lim f(x), as x approaches a number, a fraction, pi or ±∞ — two-sided, from the left or from the right. Exact values come from symbolic computation and are checked numerically, and it recognises limits that do not exist: unequal one-sided limits, oscillation or no real value nearby. Infinite limits are reported as ∞ or −∞.
使用说明与常见问题How-to & FAQ
「极限计算器」是什么？What is the Limit Calculator tool?
求函数的极限 lim f(x)：支持趋近于某个数、分数、pi 或 ±∞，可选双侧、左极限、右极限；符号计算给出精确值并用数值核对，能识别左右极限不相等、趋于无穷、振荡和无定义等“极限不存在”的情况。Find the limit of a function, lim f(x), as x approaches a number, a fraction, pi or ±∞ — two-sided, from the left or from the right. Exact values come from symbolic computation and are checked numerically, and it recognises limits that do not exist: unequal one-sided limits, oscillation or no real value nearby. Infinite limits are reported as ∞ or −∞.
如何使用「极限计算器」？How do I use the Limit Calculator tool?
在「极限计算器」中，按需设置「函数 f(x)」、「变量」、「趋近点 a」、「方向」等选项，然后点击「求极限」按钮即可。In the Limit Calculator: Set "Function f(x)", "Variable", "Approaches a", "Direction", then click "Find limit".
怎样判断极限不存在？How does it decide that a limit does not exist?
工具会在趋近点两侧（或在 ±10¹ 到 ±10⁷ 处）实际计算函数值：左、右极限不相等（如 |x|/x 在 0 处）、一侧趋于 +∞ 另一侧趋于 −∞（如 1/x）、函数不停振荡（如 sin(1/x)）或一侧没有实数值（如 √x 在 0 的左侧）时，都会显示“不存在”并说明原因。The function is actually evaluated on both sides of the point (or at ±10¹ to ±10⁷ for infinity). If the one-sided limits differ (|x|/x at 0), one side goes to +∞ and the other to −∞ (1/x), the function keeps oscillating (sin(1/x)) or one side has no real value (√x left of 0), the result is "does not exist" with the reason.
结果是精确值还是近似值？Is the result exact or approximate?
先用 nerdamer 做符号计算，再用数值计算核对：两者一致时显示精确值（如 1/2、e），并附上小数；符号计算失败或与数值不一致时，改为显示数值估计（标注“≈”，约 6 位以上有效数字可靠）。nerdamer computes the limit symbolically, then a numeric check verifies it. When they agree you get the exact value (such as 1/2 or e) with its decimal; when symbolic computation fails or disagrees, a numeric estimate marked ≈ is shown instead, reliable to about 6 or more significant digits.
怎么输入无穷大？How do I enter infinity?
在“趋近点 a”中填写 inf、∞ 或 infinity，负无穷填写 -inf 或 -∞。趋向无穷时不区分左右，“方向”选项会自动停用。Type inf, ∞ or infinity in Approaches a, and -inf or -∞ for negative infinity. Direction doesn't apply at infinity, so that option is disabled automatically.
数值估计有什么局限？What are the limits of the numeric estimate?
数值核对只在有限个点取值，对于收敛极慢的函数或在 10⁻⁵ 以内才出现变化的函数可能判断错误；遇到这类情况，建议把结果当作参考并用化简、洛必达法则等方法确认。The numeric check samples a finite set of points, so it can misjudge functions that converge extremely slowly or only change within 10⁻⁵ of the point. Treat such results as a guide and confirm them by simplifying or with L'Hôpital's rule.
支持哪些写法？What syntax is supported?
四则运算 + - * /、乘方 ^、括号，函数 sin cos tan asin acos atan sinh cosh tanh sqrt abs exp log（自然对数，ln 也可以）等，常数 pi 和 e。乘号可以省略（2x、3sin(x)），也可以用 ×、÷、π、√。+ - * /, powers with ^, parentheses, functions such as sin cos tan asin acos atan sinh cosh tanh sqrt abs exp log (natural log; ln also works), and the constants pi and e. Multiplication can be implicit (2x, 3sin(x)); ×, ÷, π and √ are accepted too.
用的是什么计算引擎？What engine does it use?
开源符号计算库 nerdamer（Apache-2.0 许可），在独立的后台线程（Web Worker）中运行，超过 10 秒会自动停止，不会卡住页面；所有计算都在本地完成。The open-source symbolic library nerdamer (Apache-2.0), running in a background Web Worker that stops after 10 seconds so the page never freezes; everything is computed locally.
计算精度如何？How precise are the results?
「极限计算器」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Limit Calculator uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「极限计算器」免费吗？需要注册吗？Is the Limit Calculator free? Do I need an account?
「极限计算器」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Limit Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.