Mastering Trigonometric Identities: Solving The Limit Of (1-Cos(X))/X As X Approaches 0

Continuity Rule

If the limit exists (aka left limit and right limit are equal), and the limit equals the function at that point.

The continuity rule is a mathematical concept that refers to the behavior of a function as its input values approach a particular point. More specifically, the continuity of a function at a given point means that there are no abrupt changes or breaks in the function’s value as the input approaches that point.

To formally define the continuity of a function, we can use the epsilon-delta definition. This definition states that a function f(x) is continuous at a point c if, for every epsilon greater than 0, there exists a delta greater than 0 such that |f(x) – f(c)| < epsilon whenever 0 < |x-c| < delta. In simpler terms, this means that if we pick any small number (epsilon) that we'd like the function to be accurate to, we can find a small interval around the point in question (c) such that the function never strays too far from its value at c inside that interval. This ensures that the function is smooth at that point, meaning there are no sudden jumps or holes. The continuity rule has important implications in calculus and other areas of mathematics. For example, in calculus, a function must be continuous to be differentiable (able to find the derivative) at a point. Additionally, many real-world phenomena can be modeled using continuous functions, making the study of continuity important in a variety of fields.

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