Proving Limits Using Delta Epsilon Method

In summary, The goal is to use the delta epsilon method to prove that the limit of x^3+2x^2-3x+4 as x approaches 1 is equal to 4. This involves finding a suitable delta value such that whenever the absolute value of x-1 is less than delta, the absolute value of the expression x^3+2x^2-3x+4-4 is less than epsilon. The steps to follow include breaking the expression into smaller parts, finding a bound for each part, and putting it all together to form a proof. It is important to be cautious and choose an appropriate delta value, such as the minimum between epsilon/10 and 1, in order to ensure the
  • #1
gaborfk
53
0

Homework Statement


Using a delta epsilon method prove:
[tex]\mathop {\lim }\limits_{x \to 1 } x^3+2x^2-3x+4= 4[/tex]


The Attempt at a Solution


I got so far as breaking the equation into

[tex]=|x||x+3||x-1|[/tex] now how do I bound it? Also, even more basic question, once I found the bound how do I put the whole thing together as a proof? I do not want you to prove this one, but please, if you can provide a link to a similar problem for me to see how it flows from idea, to figuring out the bounding, to the scratch work of figuring out epsilon, to the actual proof write up. Like the whole flow of things.

I am very familiar with "standard" proofs, but very much lost on the real analysis ones... We have a very bad book with no examples but mainly some ideas, definitions and a ton of homework.


Thank you in advance
 
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  • #2
well you need to show that

[tex]\forall\epsilon>0,\exists\delta(\epsilon)>0, \ \ such \ \ \ that \ \ \ whenever \ \ \ \\ \ \

0<|x-1|<\delta => |x^{3}+2x^2-3x+4-4|<\epsilon ?[/tex]

Now from here we go:

[tex]|x^{3}+2x^2-3x+4-4|=|x^{3}+2x^2-3x|=|x(x^{2}+2x-3)|=|x||x+3||x-1|<|x||x+3|\delta<|x+3|2\delta<10\delta=\epsilon=>\delta=\frac{\epsilon}{10}[/tex]

lets see, since x-->1 , it is safe to assume that

[tex]0<x<2=>|x|<2[/tex] also [tex]0<x<2/+3 => 3<x+3<5=>|x+3|<5[/tex]
 
  • #3
In order to be completely "safe" one should say something like [tex]\delta[/itex]= minimum of [itex]\left{\frac{\epsilon}{10}, 1\right}[/tex].
 
  • #4
HallsofIvy said:
In order to be completely "safe" one should say something like [tex]\delta[/itex]= minimum of [itex]\left{\frac{\epsilon}{10}, 1\right}[/tex].


Yes, i forgot to add that!
 

FAQ: Proving Limits Using Delta Epsilon Method

What is real analysis?

Real analysis is a branch of mathematics that deals with the study of real numbers and their properties. It involves the use of rigorous mathematical proofs to understand the behavior of real numbers, functions, and mathematical concepts.

What is a proof in real analysis?

A proof in real analysis is a logical and systematic argument that uses mathematical principles and definitions to show the validity of a mathematical statement or theorem. It is a way of demonstrating that a statement is true beyond any doubt.

What are some common techniques used in real analysis proofs?

Some common techniques used in real analysis proofs include direct proof, proof by contradiction, proof by induction, and proof by contrapositive. Other techniques may include using mathematical identities, inequalities, and limits to show the validity of a statement.

How can I improve my skills in writing real analysis proofs?

The best way to improve your skills in writing real analysis proofs is to practice regularly. Start with simple problems and gradually move on to more complex ones. It is also helpful to study and understand the proofs written by experts in the field and to seek feedback from others on your proofs.

Are there any common pitfalls to avoid when writing real analysis proofs?

Yes, there are some common pitfalls to avoid when writing real analysis proofs. These include making incorrect assumptions, using vague or imprecise language, and skipping steps without proper justification. It is important to be thorough and precise in your reasoning to avoid these pitfalls.

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