TheRedDevil18 said:Homework Statement
lim x→2 (x2+1) = 5
Homework Equations
0 < |x-a| < delta
|f(x) - L| < ε
The Attempt at a Solution
0 < |x-2| < delta
|x2-4| < ε
|(x-2) (x+2)| < ε
|x-2| |x+2| < ε.....and I am stuck here, any help
The precise definition of the limit is a mathematical concept that describes the behavior of a function as the input values approach a specific point. It states that the limit of a function f(x) as x approaches a certain value c is equal to some constant L if for any positive number ε, there exists a positive number δ such that if the distance between x and c is less than δ, then the distance between f(x) and L is less than ε.
The precise definition of the limit is a rigorous and mathematical way of describing the concept of a limit, whereas the intuitive understanding is based on general ideas and approximations. The precise definition requires a specific value to be approached and quantifies the distance between the input and output values, while the intuitive understanding may rely on visualizations or estimates.
The precise definition of the limit is crucial in various areas of mathematics, such as calculus, analysis, and differential equations. It allows for the precise calculation of limits and the evaluation of functions at specific points, which is essential in understanding the behavior of functions and solving mathematical problems.
To prove using the precise definition of the limit, you must show that for any ε > 0, there exists a δ > 0 such that if the distance between x and c is less than δ, then the distance between f(x) and L is less than ε. This can be done by manipulating the equation and using the properties of limits, such as the squeeze theorem or the algebraic limit theorem.
One common mistake when using the precise definition of the limit is incorrectly identifying the values of ε and δ and their relationship to each other. Another mistake is not considering the direction of the limit, whether it is approaching from the left or the right. It is also important to check the continuity of the function at the limit point, as this can affect the validity of the proof.