Proving the Limit of sin(x)/x = 1 using the Squeeze Theorem

This GeoGebra applet visually demonstrates the proof of the fundamental trigonometric limit: lim(x->0) sin(x)/x = 1, using the Squeeze Theorem. By constructing geometric elements such as segments and areas within a unit circle, it establishes the inequalities between sin(x), x, and tan(x). This geometric approach provides an intuitive understanding of the Squeeze Theorem and the rigorous proof of one of the most important limits in differential calculus. Students interact with a dynamic unit circle diagram where they can drag a point to vary the angle x approaching zero. As x changes, the applet highlights three key geometric quantities: the vertical segment representing sin(x), the arc length representing x, and the tangent segment representing tan(x). These visual elements are color-coded for clarity, and their relative lengths are compared in real time. The applet simultaneously displays the derived inequality sin(x) < x < tan(x) for small positive values of x, and through algebraic manipulation, guides the learner through the steps that lead to cos(x) < sin(x)/x < 1. As the angle approaches zero, the bounding functions cos(x) and 1 both converge to 1, visually confirming that sin(x)/x is squeezed between them and must therefore also approach 1. The applet further enhances comprehension by plotting the function sin(x)/x alongside its limit value of 1 on a coordinate graph. Users can observe how the curve of sin(x)/x approaches the horizontal line y = 1 from both sides as x gets arbitrarily close to zero, even though the function is undefined at exactly x = 0. This dual presentation—geometric proof within the unit circle and graphical confirmation on the Cartesian plane—reinforces the connection between abstract algebraic reasoning and concrete visual intuition. Designed for high school and college-level calculus students, this interactive tool bridges the gap between informal geometric insight and formal epsilon-delta rigor. It allows learners at different levels to engage meaningfully: beginners gain a visual grasp of why the limit equals 1, while more advanced students can appreciate how the geometric construction underpins the formal statement of the Squeeze Theorem. The applet is particularly valuable for classrooms where instructors wish to move beyond rote memorization and help students internalize the logical structure behind one of calculus foundational results.