Proving Limit Def. for Positive A & B: Help Needed!

In summary, to prove that if the limit of a function f(x) as x approaches c is equal to L, there exist positive numbers A and B such that if the absolute value of x-c is less than A, then the absolute value of f(x) is less than B. This can be proven by using the definition of a limit, where for any positive epsilon, there exists a positive delta that satisfies the condition. By setting A as delta and B as any value greater than L, the statement can be proven.
  • #1
javi438
15
0

Homework Statement



prove that if [tex]lim_{x\rightarrow c}[/tex] f(x) = L, then there are positive numbers A and B such that if 0 < |x-c|< A, then |f(x)|< B

2. The attempt at a solution

i know it's something to do with the limit definition, where for [tex]\epsilon[/tex] > 0, there exists a [tex]\delta[/tex] > 0 such that 0 < |x-c| < [tex]\delta[/tex], then |f(x)-L| < [tex]\epsilon[/tex]

i don't know how to get my way through proving it!
please helpppp!
 
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  • #2
yes, precisely, use the definition of a limit. Your task is to identify what delta and epsilon will make it works
 
  • #3
Pick a B>L. Pick epsilon=B-L. Use the definition of limit to find a delta. Set A=delta.
 

1. What is the limit definition for positive A and B?

The limit definition for positive A and B is a mathematical concept that describes the behavior of a function as the input approaches a certain value. In this case, A and B represent positive real numbers.

2. How do you prove the limit definition for positive A and B?

To prove the limit definition for positive A and B, you must show that for any positive number ε, there exists a positive number δ such that if the distance between the input and the limit value is less than δ, then the distance between the output and the limit value is less than ε.

3. What is the importance of proving the limit definition for positive A and B?

Proving the limit definition for positive A and B is important because it is the foundation for understanding and using calculus, as it allows us to rigorously define and calculate limits which are essential in many areas of mathematics and science.

4. Can the limit definition for positive A and B be used for all functions?

Yes, the limit definition for positive A and B can be used for all functions, as long as the input and output are real numbers. This definition applies to both continuous and discontinuous functions.

5. Are there any alternative ways to prove the limit definition for positive A and B?

Yes, there are alternative ways to prove the limit definition for positive A and B, such as using the epsilon-delta definition or using the squeeze theorem. However, the basic concept of showing that the output approaches a certain value as the input approaches another value remains the same.

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