# Prove using the precise definition of the limit

• TheRedDevil18
In summary, the attempt at a solution involves using the definition of a limit to find the value of delta that satisfies the given epsilon, and to manipulate the expression in such a way that the denominator does not approach 0 as x approaches 2.
TheRedDevil18

## 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

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

Continuing your analysis:$$|x-2|<\frac \epsilon {|x+2|}$$You are trying to get the right side small by getting ##x## close to ##2##. So you don't want the denominator to get close to ##0##. Can you keep it away from ##0## if ##x## is close to ##2##? That's the idea you need to pursue.

## 1. What is the precise definition of the limit?

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.

## 2. How is the precise definition of the limit different from the intuitive understanding of a limit?

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.

## 3. What is the importance of the precise definition of the limit in mathematics?

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.

## 4. How do you prove using the precise definition of the limit?

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.

## 5. What are some common mistakes when using the precise definition of the limit?

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.

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