Numerical calculus on any f(x) the site expression engine understands: derivatives via central differences (with the second derivative), definite integrals via composite Gauss–Legendre quadrature with a same-budget two-rules error estimate, and two-sided limits probed with shrinking steps.
Everything is numeric — no symbolic rewriting — which is exactly right for checking a model or sanity-checking homework: fast, honest about error, and never claims an exact form it cannot prove.
Frequently asked questions
How accurate is the derivative?
Central differences are O(h²) accurate; the step is scaled to |x₀| near the floating-point sweet spot, giving ~10 significant digits for smooth functions. Derivatives that oscillate or vanish nearby degrade gracefully to fewer digits.
What if the limit does not exist?
The table shows f(x₀±h) for h down to 10⁻⁸; sin(1/x)-style oscillation shows up as two columns that never agree, with a verdict of "no common limit".
中文说明
对站内表达式引擎支持的任意 f(x) 做数值微积分：中心差分求导（含二阶导）、复合高斯-勒让德求积（附同预算双规则误差估计）、递减步长探测双侧极限。
全部为数值方法——不做符号改写——恰好适合检验模型或核对作业：快、对误差诚实，绝不冒称求出了无法证明的解析形式。
常见问题
导数有多准？
中心差分精度 O(h²)，步长按 |x₀| 缩放到浮点最优点，光滑函数约 10 位有效数字。在 x₀ 附近振荡或趋零的函数会自然退化为较少位数。
极限不存在时会怎样？
表格给出 h 递减到 10⁻⁸ 的 f(x₀±h)；sin(1/x) 型振荡表现为两列始终不合拢，结论显示「两侧无公共极限」。