A limit is a mathematical concept that represents the behavior of a function as the input values approach a certain value or point. It is used to determine the value that a function approaches as the input values get closer and closer to a specific point.
To solve for limits algebraically, you can use the limit laws, which include the sum, difference, product, and quotient rules. These rules allow you to manipulate the function algebraically until you can evaluate the limit at the specified point.
A discontinuity is a point in a function where there is a break or a gap in the graph. It occurs when the function is not continuous at that point, meaning there is a sudden change or jump in the values of the function.
You can identify a discontinuity by looking at the graph of the function. Discontinuities can be classified as removable or non-removable. A removable discontinuity can be identified by a hole or a point where the function is undefined. A non-removable discontinuity can be identified by a jump or a vertical asymptote in the graph.
The process for solving limits and discontinuities involves identifying the type of discontinuity, if any, and using the appropriate techniques to evaluate the limit at the specified point. This may involve factoring, simplifying, or using limit laws. If the limit is indeterminate, you may need to use advanced techniques such as L'Hospital's rule. It is important to remember that the limit and the function value may not always be the same at a discontinuity.