step1536 said:Use the given graph of(x) =x^2 |x^2-1| < 1/2 whenever |x-1| <delta .The Given points on the graph are(0.8,0.5), (1.2,1.5). Please give your answer to the value of delta, where deltaor any smaller positive number will satisfy all conditions. correct to four decimals, round down if necessary.
The precise definition of a limit is a mathematical concept used to describe the behavior of a function as its input approaches a certain value. It states that the limit of a function as x approaches a certain value, say a, is equal to L if and only if for any given positive number ε, there exists a positive number δ such that if |x-a| < δ, then |f(x)-L| < ε.
The precise definition of a limit is a mathematical statement that is based on the concept of distance and uses the concept of limits of sequences to define the behavior of a function. It is a rigorous and formal definition, while the intuitive understanding of a limit is a more general idea that describes the behavior of a function at a given point.
In order for a limit to exist, the function must be defined at the point in question, the function must approach a single value as the input approaches the given value, and the behavior of the function must be consistent from both the left and right sides of the given value.
The precise definition of a limit is important because it allows us to make precise and accurate statements about the behavior of a function at a specific point. It also provides a framework for understanding and solving more complex mathematical problems involving limits.
The precise definition of a limit is used in many real-world applications, such as in the fields of physics, engineering, and economics. It can be used to analyze the behavior of systems and make predictions based on mathematical models. For example, in physics, the precise definition of a limit is used to describe the behavior of particles in motion and to calculate their velocity and acceleration at a specific point in time.