Use limit definition to prove

In summary, the conversation discusses the use of the limit definition to prove a given limit and the process of finding a suitable delta value for a given epsilon. The conversation also touches upon the idea of there being multiple correct answers for this type of problem and the importance of understanding the concept rather than just relying on a textbook's answer.
  • #1
John O' Meara
330
0
Ues the limit definition to prove that the stated limit is correct. [tex] \lim_{x->-2} \frac{1}{x+1}=-1[/tex]. The limit def' is |f(x)-L|<epsilon if 0<|x-a|< delta. So we have [tex]|\frac{1}{x+1} + 1| < \epsilon if 0 < |x- (-2)| < \delta \mbox{ therefore } |\frac{1}{x+1}||x+2| < \epslion \mbox{ if } 0 < |x + 2| < \delta[/tex]. To bound [tex] \frac{1}{x+1} \mbox { let } -1 < \delta < 1 [/tex].[tex] -1 < x+2< 1 \mbox{ that implies } -2 < x+1 < 0 \mbox{ therefore } \frac{-1}{2} > \frac{1}{x+1} > 0[/tex]. This cannot be correct as the answer for delta is [tex] \frac{\epsilon}{1+\epsilon}[/tex]. My algebra is rusty. Please help, thanks.
 
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  • #2
Not sure why you abandoned your initial line of thought, since there is no single correct answer for delta. The only problem with your analysis is that you have -2 < x + 1 < 0 and then took reciprocals, but you have 0 on one side so following the normal operations of taking reciprocals of inequalities would yield 1/(x+1) > 1/0, which is absurd. Instead, suppose you had picked delta [itex]\leq[/itex] 1/2. Then all that would change is that you now have -1/2 < x + 2 < 1/2 so -3/2 < x + 1 < -1/2, which means that 1/2 < |x+1| < 3/2 and finally 2/3 < 1/|x+1| < 2. Now you have an upper bound on 1/|x+1| without having to worry about division by zero.
 
  • #3
Using snipez90's suggestion for delta, I get [tex] \delta=min(\frac{1}{2},\frac{\epsilon}{2}) [/tex]. I was wondering how the book got the result I said it got, that is why I abandoned my original line of thought.
 
  • #4
John:

When it comes to estimates and inequalities like this, you are NOT to reproduce the "answer" the book gives you (I find it scandalous and deeply anti-pedagogic that a textbook actually provides "answers" to such exercises!)

Depending on how you think, numerous CORRECT deductions might yield very different delta-values, as functions of the epsilon.

It can be that we may find the "optimal" (i.e, largest) delta that CAN work, but this does not in any way change the validity of a deduction leading to a smaller delta.

Thus, there is no single, correct answer to these types of questions (there are many correct answers)
 
  • #5
I am not very familiar with the delta epsilon proof , but is it not sufficient to show that for a given delta you can find an epsilon , do we need to show the opposite ?
 
  • #6
I am sorry if I broke one of your rules. It was not my intension to do so. As I am teaching myself from a University maths book, which are generally not designed with the self taught in mind. I find that the answer to odd numbered questions is in general the only guide as to how I might be progressing. I admit that that is not ideal and that having some answers to questions can throw you (as it did here) off a valid line of reasoning or investigation. All I can do is offer my apology again.

I am just learning the epsilon delta proof myself. Thanks for the replies.
 

1. What is the definition of a limit?

The limit of a function at a certain point is the value that the function approaches as the input approaches that point. It represents the behavior of the function near that point.

2. How do you prove a limit using the limit definition?

To prove a limit using the limit definition, you must show that for any given small positive number, there exists a small positive number such that when the input is within that distance from the point, the output is within the given small positive number from the limit.

3. What are the steps for using the limit definition to prove a limit?

The steps for using the limit definition to prove a limit are as follows: 1. Write out the definition of a limit using the given function and point. 2. Simplify the expression using algebraic techniques. 3. Use the given small positive number to find a corresponding small positive number. 4. Substitute this number into the expression to show that the distance between the output and the limit is within the given small positive number.

4. Can you use the limit definition to prove all limits?

Yes, the limit definition can be used to prove all limits. However, for some limits, it may be more efficient to use other methods such as algebraic techniques or theorems.

5. Are there any common mistakes when using the limit definition to prove a limit?

Yes, some common mistakes when using the limit definition to prove a limit include not following the steps correctly, not choosing an appropriate small positive number, and not simplifying the expression enough. It is important to be careful and thorough when using the limit definition to avoid these mistakes.

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