A limit is the value that a function approaches as its input (x) approaches a specific value or "approach point" (a). It is denoted by the notation "lim x→a" and is used to describe the behavior of a function near a certain point.
To solve for a limit, you can use a variety of methods such as direct substitution, factoring, and algebraic manipulation. You can also use graphs or tables to estimate the limit. It is important to consider the behavior of the function as x approaches the given value to determine the limit value.
Finding a limit helps us understand the behavior of a function near a specific point. It also allows us to evaluate functions that are not defined at a certain point, as long as the limit exists.
A limit exists if the left and right-hand limits are equal. This means that the function approaches the same value from both directions. Additionally, the limit exists if the function is continuous at the given point, meaning there are no jumps or breaks in the graph.
The limit laws state that for two functions, f(x) and g(x), the limit of their sum, difference, product, or quotient can be found by taking the limit of each individual function and then applying the corresponding operation. For example, for the given function, you can split it into two separate limits: lim(x→8) (x-8) and lim(x→8) (x^(1/3)-2). Then you can use substitution to evaluate each individual limit and apply the division rule to find the overall limit.