This demo shows two blue lines through a point (x₀, f(x₀)), one with a slope ε greater than that of the tangent line and one with a slope ε less than that of the tangent line. If the function is differentiable at x₀, you should always be able to choose a δ (controlled by the hotspot) small enough that the portion of the curve between x₀ - δ and x₀ + δ lies inside the blue lines.