x-2/x-3 = 1 + 1/ x-3
When taking a limit, we are essentially finding the value that a function approaches as its input approaches a certain value. This value is known as the limit and is often denoted by the notation lim x→a f(x).
Taking limits allows us to understand the behavior of a function near a specific point, even if the function may not be defined at that point. This is useful in many mathematical and scientific applications, such as analyzing the rate of change of a quantity or determining the convergence of a series.
There are several common techniques for evaluating limits, including direct substitution, factoring, rationalization, and the use of special limit rules such as the limit laws and L'Hôpital's rule. These techniques can be used to simplify the expression and determine the limit more easily.
A limit exists if the values of the function approach the same value from both the left and right sides of the limit point. In other words, the left-hand limit and the right-hand limit must be equal. If this condition is met, then the limit exists and is equal to this common value.
Some common types of problems involving limits include indeterminate forms, infinite limits, and limits at infinity. These types of problems often require specific techniques and strategies to solve, such as using algebraic manipulation, graphing, or trigonometric identities.