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 pertains to the study of limits in calculus. It is a property that states that the limit of a function exists and equals the function value at a point if the function is continuous at that point.
In simpler terms, if you can draw a function without lifting your pencil from the paper, then the function is continuous. This means that the function has no holes, jumps, or breaks in it.
The continuity rule is important in calculus because it allows us to evaluate limits at certain points without having to use complicated algebraic techniques. This is because, if a function is continuous at a certain point, we can simply plug in that point into the function to find its limit.
For example, consider the function f(x) = x^2. This function is continuous for all values of x, meaning that its limit at any point exists and equals its function value at that point. So, if we want to find the limit of f(x) as x approaches 2, we simply plug in 2 for x to get f(2) = 4. Therefore, the limit of f(x) as x approaches 2 is 4.
In summary, the continuity rule is a crucial property in calculus that allows us to evaluate limits more easily by determining if a function is continuous at a particular point.
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