Using delta epsilon. inequality manipulation

In summary, the conversation discusses the challenges of proving limits of multivariable functions, specifically using the delta-epsilon proof method. The speaker is struggling with setting up the inequalities and manipulating them to find delta in terms of epsilon. They are also concerned about their ability to do this for their future career in math/physics. The conversation also mentions the importance of being proficient in single variable limits before attempting multivariable limits.
  • #1
yaje
9
0
Im having trouble proving limits of multivariable functions. I understand the principal behind delta-epsilon proofs but I can't get it to work. Once I set up the inequalities I am stuck.

The only example in my book seems very convenient though .

3x^2 * abs(y) divided by x^2 + y^2 It uses x^2 divided by x^2 +y^2 is less than or equal to one so (3x^2 * abs(y) divided by x^2 + y^2) is less than or equal to 3 abs(y).

Im guessing that you are supposed to manipulate equalitys to find delta in terms of epsilon in all cases, is that right?

It may be that I don't have the intuitive sense to just synthesize the proper way to do this "manipulation" in my homework problems. I understood the method the book used but it would not have been clear had it not been shown step by step. Id like to find some sources that would help me get a better feel for this process so I came here asking for help.

Also as someone who is looking forward to a carrer in math/physics how concerned should I be that I am unable to just DO this? The book assumes that I can get delta in terms of epsilon. I can't so I am concerned.
 
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  • #2
yaje said:
Im having trouble proving limits of multivariable functions. I understand the principal behind delta-epsilon proofs but I can't get it to work. Once I set up the inequalities I am stuck.

The only example in my book seems very convenient though .

3x^2 * abs(y) divided by x^2 + y^2 It uses x^2 divided by x^2 +y^2 is less than or equal to one so (3x^2 * abs(y) divided by x^2 + y^2) is less than or equal to 3 abs(y).

Im guessing that you are supposed to manipulate equalitys to find delta in terms of epsilon in all cases, is that right?

It may be that I don't have the intuitive sense to just synthesize the proper way to do this "manipulation" in my homework problems. I understood the method the book used but it would not have been clear had it not been shown step by step. Id like to find some sources that would help me get a better feel for this process so I came here asking for help.

Also as someone who is looking forward to a carrer in math/physics how concerned should I be that I am unable to just DO this? The book assumes that I can get delta in terms of epsilon. I can't so I am concerned.

yaje:

Before you get involved with multi variable limits you should check yourself how good you are with a single variable limits.

Can you for example do the following limits quite easily:

1) [tex]lim_{x\to 1}{2x+3} = 5[/tex]

2)[tex] lim_{x\to 1}x^2 = 1[/tex]

3) [tex] lim_{x\to 2}\sqrt x =\sqrt 2[/tex]

4)[tex] lim_{x\rightarrow 0} sinx = 0[/tex]
 

Related to Using delta epsilon. inequality manipulation

What is delta-epsilon inequality manipulation?

Delta-epsilon inequality manipulation is a technique used in mathematical analysis to prove the limit of a function. It involves manipulating the delta-epsilon definition of a limit to show that, for any given epsilon, there exists a corresponding delta such that the function's values are within epsilon of the limit whenever the function's input values are within delta of a particular point.

Why is delta-epsilon inequality manipulation important?

Delta-epsilon inequality manipulation is important because it provides a rigorous and precise method for proving the limit of a function. This is essential in many areas of science, such as physics and engineering, where accurate predictions and calculations rely on understanding the behavior of functions at their limits.

How do you use delta-epsilon inequality manipulation?

To use delta-epsilon inequality manipulation, you first need to understand the definition of a limit and the concept of delta and epsilon. Then, you can manipulate the definition to show that for any given epsilon, there exists a corresponding delta that satisfies the definition. This typically involves using algebraic and logical manipulations to find an appropriate delta value.

What are some common mistakes when using delta-epsilon inequality manipulation?

Some common mistakes when using delta-epsilon inequality manipulation include confusing delta and epsilon values, using incorrect algebraic manipulations, and not fully understanding the definition of a limit. It is important to carefully and accurately follow the steps of the manipulation and check for errors to avoid these mistakes.

Can delta-epsilon inequality manipulation be used for all functions?

Delta-epsilon inequality manipulation can be used for many types of functions, including polynomial, exponential, and trigonometric functions. However, it may not be applicable to all functions, particularly those that are discontinuous or have infinite limits. In these cases, other methods may be necessary to prove the limit of a function.

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