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Differential Calculus2D

Theorems of Calculus - Theory & Concepts - L Hopital Rule

Understand the foundational theorems that bridge the gap between continuous functions and derivatives: the Extreme Value Theorem, Rolle's Theorem, the Mean Value Theorem, and Cauchy's MVT.

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Case 1: sin(x) / x

Explore how both curves become linear under high zoom, showing that their ratio approaches the ratio of their tangent lines.

lim⁡x→0sin⁡xx[00]\lim_{x \to 0} \frac{\sin x}{x} \quad \left[\frac{0}{0}\right]
-1.5Target c = 01.5
1x (Wide View)50x (Linear Zoom)
Limit Approximations
f(x) = \sin x0.479426
g(x) = x0.500000
Ratio f(x)/g(x)0.958851
Slopes f'(x)/g'(x)0.877583
Blue: f(x)Orange: g(x)Vertical line: Current x position
Loading chart...
As you zoom to 50x, observe how both curves look perfectly straight (tangent line approximation) and intersect exactly at y=0y = 0 at x=0x = 0.

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