It looks to me like the problem should read something like:kramer733 said:Homework Statement
if |x-3| < ε/7 and 0 < x ≤ 7 prove that |x^2 - 9| < ε
Homework Equations
The Attempt at a Solution
So ths is what I did so far.
|x+3|*|x-3| < ε (factored out the |x^2 - 9|)
|x+3|*|x-3| < |x+3|* ε/7 < ε (used the fact that |x-3| < ε/7)
|x+3|* ε/7 *7 < ε*7|x-3| < ε/7*7 (multiplied both sides of the inequality by 7)
I suck at epsilon delta proofs and have no idea where to go from here.
kramer733 said:How do i get started on this proof then/?
Correct.kramer733 said:x+3<7
An epsilon delta proof is a method used to rigorously prove the limit of a function. It involves using the concept of an arbitrarily small number (epsilon) and finding a corresponding value (delta) which, when added to or subtracted from the input of the function, results in an output that is within epsilon of the desired limit.
An epsilon delta proof is important because it allows us to prove the existence of a limit with mathematical certainty. It also helps us to understand the behavior of a function near a specific point and can be used to prove the continuity of a function.
To solve this proof, we must first define the limit we are trying to prove. In this case, we are trying to prove that the limit of x^2 as x approaches 3 is equal to 9. Next, we must manipulate the given inequality to isolate x. We can do this by factoring the expression inside the absolute value and setting it equal to ε, giving us the expression |x - 3||x + 3| = ε. Then, we can choose a value for delta, such that when we substitute it into the expression, it satisfies the inequality. Finally, we must show that for all values of x that are within delta of 3, the expression |x - 3||x + 3| will be less than ε, thereby proving the limit.
Some common mistakes when solving an epsilon delta proof include choosing an incorrect value for delta, not properly manipulating the given inequality, and not properly showing that the chosen delta satisfies the inequality for all x values within delta of the desired limit.
Epsilon delta proofs can be applied in various real-world situations, such as in engineering and physics, to prove the existence of limits and the continuity of functions. They can also be used in economics and statistics to analyze data and make predictions. In general, epsilon delta proofs can be used to provide rigorous mathematical explanations and justifications in any field that involves the use of limits and functions.