Imagine you observe a clock pendulum swinging very slowly approaching its equilibrium point. This motion is similar to the behavior of trigonometric functions when their variable approaches a certain value. Limits of trigonometric functions examine how the values of sine, cosine, and tangent functions behave when the input approaches critical points.
Unlike limits of algebraic functions that can often be solved by direct substitution, trigonometric functions have special characteristics due to their periodic and oscillating nature. This makes us need to use theorems and special properties to solve trigonometric limits.
The most important foundation in trigonometric limits is the theorem stating that the sine function approaches its gradient when the angle approaches zero.
This theorem cannot be proven using direct substitution because it results in the form 00. Its proof requires a geometric approach using the unit circle and the squeeze theorem.
In this theorem, x must be in radians, not degrees. If using degrees, the result will be different.
