Math Visualizations

What is a Limit?

A limit describes the value that a function approaches as the input approaches a specific point. Limits are the foundation of calculus.

The derivative is defined as a limit:

$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Secant vs Tangent

The secant line connects two points on a curve. Its slope gives the average rate of change between those points.

The tangent line touches the curve at one point. Its slope gives the instantaneous rate of change at that point.

As the two points get closer together, the secant line approaches the tangent line:

$$\text{slope of secant} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} \xrightarrow{t_2 \to t_1} s'(t_1)$$

Try This

1. Move the slider to change t₂ and watch the secant line
2. Bring t₂ close to t₁ = 5 — the secant slope approaches the tangent slope
3. Toggle the tangent line to compare secant and tangent directly
4. Notice that when t₂ = t₁, the secant slope equals the tangent slope (the derivative)

Real-Life Applications

• Instantaneous speed from average speed between two points on a GPS track
• Marginal cost as the limit of average cost change
• Reaction rates in chemistry as concentration change approaches zero time interval