Proving Limit lim((n+2)/(n^2-3))=0 with Definition of a Limit

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In summary, as n goes to infinity, the limit of (n+2)/(n^2-3) approaches 0. This can be shown using the definition of a limit by setting N>ε+2 and manipulating the expression to get 1/n, and then letting N>1/ε. It is also possible to make the fraction smaller by multiplying the denominator by a larger number, such as sqrt(n).
  • #1
mjjoga
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lim((n+2)/(n^2-3))=0 as n goes to infinity. I can only use the definition of a limit.

My work so far,

I'm trying to work out what n will be greater than. I have:
|(n+2)/(n^2-3)|=(n+2)/(n^2-3) if n is greater than one.
From here, I have been trying anything to get rid of the addition and subtraction.
I got 1/(n-2) but that can't work. I'm not sure how to manipulate it. I tried multiplying by n, making perfect squares. I'm stuck. If I could get a hint, that would be great.
mjjoga
 
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  • #2
mjjoga said:
|(n+2)/(n^2-3)|=(n+2)/(n^2-3) if n is greater than one.

I tried plugging in n=1.1 and got a positive number on the right and a negative number on the left. I think you mean if n>sqrt(3), unless you're only referring to integers n.
 
  • #3
hi mjjoga! :smile:

(try using the X2 icon just above the Reply box :wink:)

can you do lim (n + 2)/(n2 - 4) ? :wink:
 
  • #4
you know what, when I was canceling out the n+2 on the top, I had put down (n+2)2 in the denominator (I tried to use the x^2 button) so I left x+2 on the bottom, but it's the sum and difference so I need n-2 on the bottom. then it's fine. Sometimes my algebra just flies out the window...
Thank you bunches,
mjjoga
ps-I might be back, I've got 5 problems left...
 
  • #5
n is in natural numbers
 
  • #6
Define N>ε+2. Then whenever n>max(N,2), we have |(n+2)/(n^2-3)|<(n+2)/(n^2-3)<(n+2)/(n^2-4)=1/(n-2)<1/(ε+2-2)=1/ε
1/epsilon isn't less than epsilon. I don't know what I'm doing wrong. i added that n>2 so that the denumerator does not equal 0. I know that 1/(n-2) is right, but what else can i do to it
 
  • #7
mjjoga said:
Define N>ε+2.

erm :redface:

ε is very small, but N is very large …

wouldn't you be better off using 1/ε ? :wink:
 
  • #8
hm, but epsilon could be a tiny fraction and make 1/epsilon really big. There must be some sort of trick we can use to get it less than epsilon.
 
  • #9
mjjoga said:
… to get it less than epsilon.

you don't need N < ε, you need an M such that if N > M, then the whole thing is < ε …

your ε must be very small, and your N (or M) must be very large
 
  • #10
I figured it out. If I multiply by -1/-1 then with the different signs, I can get rid of the addition and subtraction. I did this before I even factored. then it simplifies to 1/n, so I let N>1/epsilon and it works.
 
  • #11
mjjoga said:
thanks for the help! by the way, if I know that n is greater than 1 and is a natural number, can i make a fraction bigger by multiplying the denominator by sqrt(n)?

hi myriam! :smile:

yes …

the simple rule is that if you make the denominator bigger, you make the whole thing smaller (and vice versa) :wink:
 

1. What is the definition of a limit?

The definition of a limit is a mathematical concept that describes the behavior of a function as the input approaches a certain value. In other words, it determines the value that a function approaches as its input gets closer and closer to a specific value.

2. How do you use the definition of a limit to prove a limit?

To prove a limit using the definition, you must show that for any given epsilon (ε), there exists a corresponding delta (δ) such that if the distance between the input and the limit point is less than δ, then the distance between the output and the limit value is less than ε. This can be written mathematically as: if |x-a| < δ, then |f(x)-L| < ε.

3. What is the limit of the function (n+2)/(n^2-3) as n approaches infinity?

The limit of this function as n approaches infinity is equal to 0. This can be proven using the definition of a limit by showing that for any small positive value of epsilon (ε), there exists a large enough value of n (N) such that if n>N, then |(n+2)/(n^2-3)-0| < ε.

4. Can the limit of a function be proven using a direct substitution?

No, the limit of a function cannot be proven using a direct substitution. This method only works for certain types of functions, such as polynomials. The definition of a limit must be used to prove the limit of a function in general.

5. Are there any other methods for proving a limit besides using the definition?

Yes, there are other methods for proving a limit, such as the Squeeze Theorem, the Intermediate Value Theorem, and L'Hopital's Rule. However, the definition of a limit is the most fundamental and widely used method for proving limits in calculus.

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