A limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value. It represents the value that the function "approaches" as the input gets closer and closer to the specified value.
To calculate a limit, you need to evaluate the function at values that are closer and closer to the specified value. This can be done by plugging in values that are very close to the specified value on either side, and then observing the pattern of the outputs. You can also use algebraic techniques and rules to simplify the function and determine the limit.
A one-sided limit only considers the behavior of the function as the input approaches the specified value from one side (either the left or the right). A two-sided limit considers the behavior from both sides and requires that the function approaches the same value from both sides in order for the limit to exist.
A limit exists if the function approaches a finite value as the input gets closer and closer to the specified value. If the function approaches different values from the left and the right, or if it approaches infinity, then the limit does not exist.
L'Hôpital's rule is a mathematical technique used to evaluate limits involving indeterminate forms (such as 0/0 or ∞/∞). It states that if the limit of a quotient of two functions is indeterminate, then the limit can be evaluated by taking the derivative of the numerator and denominator separately and then recalculating the limit. This rule is typically used when other algebraic techniques fail to evaluate the limit.