Continuity is a mathematical concept that describes the smoothness and unbrokenness of a function. A function is said to be continuous if there are no sudden jumps or breaks in its graph.
The formula for arctan x / x is arctan(x)/x, where arctan(x) is the inverse tangent of x.
The continuity of arctan x / x at 0 is important because it allows us to evaluate the function at that point and use it in other calculations. It also helps us understand the behavior of the function near 0.
The continuity of arctan x / x at 0 can be determined by evaluating the limit of the function as x approaches 0. If the limit exists and is equal to the function value at 0, then the function is continuous at 0.
Yes, the continuity of arctan x / x at 0 can be proven using the epsilon-delta definition of continuity. This involves showing that for any given positive number (epsilon), there exists a positive number (delta) such that if the distance between x and 0 is less than delta, then the distance between f(x) and f(0) is less than epsilon.