Proving a limit to infinity using epsilon-delta

• MHB
• brooklysuse
In summary, when looking at the limit of a function as x approaches infinity, we can use the epsilon-delta definition to prove that the limit is equal to infinity. This means that for any given number d, there exists a value k such that when x is greater than k, the function's output will always be greater than d. This can be shown by taking k to be (d-3)/2.
brooklysuse
lim 2x + 3 = ∞.
x→∞

Pretty intuitive when considering the graph of the function. But how would I show this using the epsilon-delta definition?Thanks!

Please expand the definition of the limit and write what you need to prove.

Looking to use this definition. f:A->R, A is a subset of R, (a, infinity) is a subset of A.

lim f(x) =infinity if for any d in R, there exists a k>a such that when x>k, then f(x)>d.
x->infinity

You may forget about $a$. So you have to prove that for every $d$ there exists a $k$ such that if $x>k$, then $f(x)=2x+3>d$. So consider an arbitrary $d$. You need to show that there exists a $k$ such that $x>k$ implies $x>(d-3)/2$. It's sufficient to take $k=(d-3)/2$.

1. How do you prove a limit to infinity using epsilon-delta?

The epsilon-delta proof is a method used to rigorously prove that a limit tends to infinity. It involves showing that for any arbitrarily small value of epsilon, there exists a corresponding value of delta such that the difference between the limit and the input is less than epsilon whenever the input is within delta of the limit.

2. What is the importance of using the epsilon-delta proof?

The epsilon-delta proof is important because it provides a rigorous and precise way of proving that a limit tends to infinity. It eliminates any ambiguity and ensures that the limit truly approaches infinity. It is also a fundamental concept in calculus and is used in many other proofs and theorems.

3. What are the key steps in a typical epsilon-delta proof?

The key steps in an epsilon-delta proof are: 1) Choose an arbitrary value of epsilon, 2) Use algebraic manipulation to find a corresponding value of delta, 3) Show that when the input is within delta of the limit, the difference between the limit and the input is less than epsilon.

4. Can you give an example of a limit to infinity proof using epsilon-delta?

Sure, let's consider the limit as x tends to infinity of 1/x. We want to prove that this limit is equal to 0. So, for any arbitrarily small value of epsilon, we need to find a corresponding value of delta. Let's say we choose epsilon = 0.01. Then, we can choose delta = 100, because when x > 100, the difference between 1/x and 0 is always less than 0.01. Therefore, we have proven that the limit as x tends to infinity of 1/x is equal to 0.

5. Are there any common mistakes to avoid when using the epsilon-delta proof?

Yes, there are a few common mistakes to avoid when using the epsilon-delta proof. One is assuming that delta must be a specific value, when in fact it can be any value that satisfies the given conditions. Another mistake is using the same value of delta for different values of epsilon, when in reality delta may need to change depending on the chosen epsilon. It is also important to carefully consider the given conditions and ensure that the proof is valid for all possible values of the input.

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