Prove a limit by the limit theory

In summary, the conversation discusses the application of the limit definition to prove the limit of a function. The key steps involve choosing an arbitrary epsilon, finding an N value that satisfies the given condition, and proving that the function gets increasingly close to the limit as N gets higher. It is also important to prove that the function is monotone increasing and satisfies the given condition for all n. Further clarification is needed for understanding the method of proving limits.
  • #1
kesun
37
0
Apply the limit definition to prove [tex]lim_{n\rightarrow\infty}\frac{n^{2}-1}{2n^{2}+3}=\frac{1}{2}[/tex]



(question stated above)



I started by writing it as |f(n) - 1/2| and attempted to reduce it, but I don't think it's reducible so I am not able to simplify it..

By looking at it further, it stuck me because I don't know where to go with this exactly. I know I am supposed to come up with this arbitrary [tex]\epsilon[/tex] then somehow prove that |f(n) - 1/2| < [tex]\epsilon[/tex]. I need to know what are the exact steps to prove stuff like this...
 
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  • #2
So we choose arbitrary [tex]\epsilon > 0[/tex]. This means that [tex]\epsilon[/tex] can be ANYTHING positive. Now, given this [tex]\epsilon[/tex], we want to find an [tex]N \in \mathbb{N}[/tex] such that for all [tex]n>N,\, |f(n)-1/2|<\epsilon[/tex], that is, [tex]|\frac{n^2-1}{2n^2+3}-1/2|<\epsilon.[/tex] Now, fortunately for us, our function is monotone increasing, so all we have to find is a number N satisfying [tex]\frac{N^2-1}{2N^2+3} > 1/2-\epsilon[/tex]. I think you can do the rest. (You also need to prove that [tex]1/2 > \frac{n^2-1}{2n^2+3}[/tex] for all n for this proof to work.)

The concept of the proof is as follows. Since [tex]\epsilon>0[/tex] can be made as small as we want, for ANY given small number, we can find a number N so high that for every number greater than N, the difference between 1/2 and the value of the function is smaller than the small number [tex]\epsilon[/tex]. This just says that as N gets increasingly higher, the value of the function gets increasingly close to 1/2.
 
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  • #3
phreak said:
So we choose arbitrary [tex]\epsilon > 0[/tex]. This means that [tex]\epsilon[/tex] can be ANYTHING positive. Now, given this [tex]\epsilon[/tex], we want to find an [tex]N \in \mathbb{N}[/tex] such that for all [tex]n>N,\, |f(n)-1/2|<\epsilon[/tex], that is, [tex]|\frac{n^2-1}{2n^2+3}-1/2|<\epsilon.[/tex] Now, fortunately for us, our function is monotone increasing, so all we have to find is a number N satisfying [tex]\frac{N^2-1}{2N^2+3} > 1/2-\epsilon[/tex]. I think you can do the rest. (You also need to prove that [tex]1/2 > \frac{n^2-1}{2n^2+3}[/tex] for all n for this proof to work.)

The concept of the proof is as follows. Since [tex]\epsilon>0[/tex] can be made as small as we want, for ANY given small number, we can find a number N so high that for every number greater than N, the difference between 1/2 and the value of the function is smaller than the small number [tex]\epsilon[/tex]. This just says that as N gets increasingly higher, the value of the function gets increasingly close to 1/2.

I get what you have said and proved that [tex]1/2 > \frac{n^2-1}{2n^2+3}[/tex] for all n, but I am still not very sure when you said to find an N, do you mean any arbitrary N or is it found by some sophistic method? I am not very getting used to this method of proving limit yet, so I might need a bit more details and a few key steps.. :)
 

1. What is the limit theory?

The limit theory is a mathematical concept used to prove the value of a limit by finding the behavior of a function as the input approaches a specific value.

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

To prove a limit using the limit theory, you need to show that the function approaches a specific value as the input gets closer and closer to a given value. This can be done through algebraic manipulation, substitution, or graph analysis.

3. What is the difference between a one-sided and two-sided limit?

A one-sided limit only considers the behavior of a function from one side of a given value, while a two-sided limit takes into account both sides. One-sided limits are used when the function has a discontinuity or a vertical asymptote at the given value.

4. Can a limit exist without the function being defined at the given value?

Yes, a limit can exist even if the function is not defined at the given value. This is because the limit only looks at the behavior of the function as the input approaches the given value, not the actual value at that point.

5. Are there any limitations to using the limit theory to prove a limit?

Yes, the limit theory may not work for all functions. Some functions may have more complex behavior that cannot be determined by simply looking at the input approaching a given value. In these cases, other methods such as L'Hospital's rule may be used to evaluate the limit.

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