Epsilon-delta proof of one sided infinite limit.

In summary, to prove the limit \lim_{x\rightarrow 1^+}\frac{1}{(x-1)(x-2)}=-∞, we need to find a δ>0 such that for every N<0, when 0<x-1<δ, the expression \frac{1}{(x-1)(x-2)} is less than N. By choosing δ=1 and assuming 0<x-1<-\frac{1}{N}, we can show that \left(-\frac{1}{x-1}\right)\left(-\frac{1}{x-2}\right) < N\left(-\frac{1}{x-2}\right), thus
  • #1
reinloch
5
0

Homework Statement


proof this limit:
[itex]\lim_{x\rightarrow 1^+}\frac{1}{(x-1)(x-2)}=-∞[/itex]


Homework Equations





The Attempt at a Solution



So for every [itex]N < 0[/itex], I need to find a [itex]\delta > 0[/itex] such that
[itex]0 < x - 1 < \delta \Rightarrow \frac{1}{(x-1)(x-2)} < N[/itex]

Assuming [itex]0 < x - 1 < 1[/itex], I get [itex]-1 < x - 2 < 0[/itex], and [itex]-\frac{1}{x-2}>1[/itex].

Assuming [itex]0 < x - 1 < -\frac{1}{N}[/itex], I get [itex]-(x-1) > \frac{1}{N}[/itex], [itex]-\frac{1}{x-1} < N[/itex], and [itex]\left(-\frac{1}{x-1}\right)\left(-\frac{1}{x-2}\right) < N\left(-\frac{1}{x-2}\right)[/itex], but then I got stuck.
 
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  • #2
welcome to pf!

hi reinloch! welcome to pf! :smile:
reinloch said:
[itex]\left(-\frac{1}{x-1}\right)\left(-\frac{1}{x-2}\right) < N\left(-\frac{1}{x-2}\right)[/itex]

the trick is to choose δ so that 1/(x - 2) is less than a fixed number :wink:
 
  • #3
Thanks. I am stuck with the right choice for [itex]\delta[/itex]. I choose 1 and [itex]-\frac{1}{N}[/itex], and it didn't seem to work.
 
  • #4
choose δ so that x doesn't get too close to 2 :wink:
 

1. What is an epsilon-delta proof of one sided infinite limit?

An epsilon-delta proof of one sided infinite limit is a mathematical method used to rigorously prove that the limit of a function as the input approaches infinity (or negative infinity) exists and has a specific value. It involves choosing a small number (epsilon) and finding a corresponding number (delta) such that when the input is within a certain distance (delta) from infinity, the output is within a certain distance (epsilon) from the desired limit. This ensures that the function approaches the limit value as the input gets closer and closer to infinity.

2. Why is an epsilon-delta proof necessary for proving one sided infinite limits?

An epsilon-delta proof is necessary for proving one sided infinite limits because it provides a rigorous and precise way to show that a limit exists and has a specific value. It also allows for a clear understanding of how the function behaves as the input approaches infinity or negative infinity.

3. How is an epsilon-delta proof different from other methods of proving limits?

An epsilon-delta proof is different from other methods of proving limits because it uses a specific and systematic approach to show that the limit exists and has a specific value. Other methods, such as the squeeze theorem or direct substitution, may be quicker and easier to use, but they do not provide the same level of rigor and precision as an epsilon-delta proof.

4. What are the key steps in an epsilon-delta proof of one sided infinite limit?

The key steps in an epsilon-delta proof of one sided infinite limit are:

  • Start by stating the limit definition and identifying the limit value.
  • Choose a small number (epsilon) and express it as a function of delta.
  • Find an expression for the function that is within epsilon distance from the limit value when the input is within delta distance from infinity or negative infinity.
  • Solve for delta in terms of epsilon, usually by setting the function expression equal to epsilon and solving for delta.
  • Choose a specific value for delta that satisfies the equation and show that the function is within epsilon distance from the limit value for all inputs within delta distance from infinity or negative infinity.

5. How can an epsilon-delta proof of one sided infinite limit be applied to real-life situations?

An epsilon-delta proof of one sided infinite limit can be applied to real-life situations in various fields, such as physics, engineering, and economics. It can be used to determine the behavior of functions as the input approaches infinity or negative infinity, which can be useful for predicting future trends or making decisions based on mathematical models. For example, it can be used to analyze the growth rate of a population, the decay of a radioactive substance, or the convergence of a financial investment.

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