Solving the Limit of a Sequence: 5n^2/(n^2+2)

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In summary, the limit of the sequence {an} = 5n^2 / (n^2 + 2) can be found by factoring out the constant and then dividing the top and bottom by n^2. This results in an indeterminate form of infinity/infinity, which can have a value of 0, infinity, or anything in between. The aim with limits is to find which term becomes negligible as n gets larger and larger and drop it. In this case, the 2 is ignored and the limit can be evaluated using L'Hospital's rule.
  • #1
physics=world
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1. Finding the limit of the sequence:

{ an } = 5n^(2) / (n^(2) + 2)





Homework Equations





3. what i did was :

lim as (n -> Infinity) of function [5n^(2) / (n^(2) + 2)]


Then factored out the constant:


5{lim as (n -> Infinity) of function [n^(2) / (n^(2) + 2)]}

so at this point i plug in infinity for the function
and this is where i need help.

how is it of the indeterminate form infinity/infinity.

when i plug it in i get infinity / (infinity + 2)

so isn't it just infinity?
 
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  • #2
hi physics=world! :smile:
physics=world said:
how is it of the indeterminate form infinity/infinity.

when i plug it in i get infinity / (infinity + 2)

so isn't it just infinity?

an indeterminate form is exactly that … indeterminate!

ie, you can't give it a value

∞/∞ can be 0 or ∞ or anything in between

hint: divide top and bottom by n2 :wink:​
 
  • #3
hmm it works when i use your hint.. dividing be n^2

but i just can't understand why it is of indeterminate form infinity/infinity

when i plug it in i get infinity / (infinity + 2) which would equal [infinity / 2] ?

so would that be just infinity?

im trying to understand it so i can use L'Hospitals rule.
 
  • #4
hi physics=world! :smile:

i don't understand this line …
physics=world said:
when i plug it in i get infinity / (infinity + 2) which would equal [infinity / 2] ?

where did the ∞ on the bottom go? :confused:

∞ is lot larger than 2 (!), so why are you ignoring it, instead of ignoring the 2 ? :wink:
 
  • #5
The aim with limits is to avoid writing [itex]n=\infty[/itex] by thinking about what happens as n gets larger and larger. Some terms become less and less significant as n grows. You describe them as 'negligible' and drop them and, if the limit is nice, the answer drops out.

Which term becomes negligible?
 
  • #6
what i was thinking was that infinity was like 0. so i just thought it would be infinity over 2.

so, the 2 is supposed to be ignored?
 
  • #7
physics=world said:
what i was thinking was that infinity was like 0

no!

∞ is as different from 0 as you can get …

a reasonably safe rule is that anything you can do with 0, you can't do with ∞ ! :smile:
so, the 2 is supposed to be ignored?

yup! :biggrin:
 
  • #8
so for example if it was say 5 / infinity

would the answer be zero? or infinity? or undefined?
 
  • #9
physics=world said:
so for example if it was say 5 / infinity

would the answer be zero? or infinity? or undefined?
only ∞/∞ is undefined

anything-else/∞ is 0 (because anything-else is negligible compared with ∞) :wink:
 

FAQ: Solving the Limit of a Sequence: 5n^2/(n^2+2)

What is the limit of the sequence 5n^2/(n^2+2)?

The limit of a sequence is the value that the terms of the sequence approach as n (the position in the sequence) approaches infinity. In this case, the limit of the sequence 5n^2/(n^2+2) is 5 because as n becomes larger and larger, the 2 in the denominator becomes negligible compared to n^2, and the expression essentially becomes 5n^2/n^2, which simplifies to 5.

How do you determine the limit of a sequence algebraically?

To determine the limit of a sequence algebraically, you can use the following steps:
1. Write out the sequence.
2. Divide each term by the highest power of n in the sequence.
3. Simplify the resulting expression.
4. Take the limit as n approaches infinity.
5. If the resulting expression is a finite number, then that is the limit of the sequence. Otherwise, the limit does not exist.

What is the significance of the limit of a sequence?

The limit of a sequence is important because it allows us to determine the behavior of the sequence as the number of terms increases. It also helps us to find the value that the sequence approaches as n approaches infinity. This can be useful in many real-world applications, such as in physics and economics.

Can the limit of a sequence be negative?

Yes, the limit of a sequence can be negative. The limit is simply the value that the terms of the sequence approach, and it can be any real number (including negative numbers). For example, the limit of the sequence (-1)^n is -1, as the terms alternate between 1 and -1 and approach -1 as n approaches infinity.

Is it possible for a sequence to have no limit?

Yes, it is possible for a sequence to have no limit. This can happen if the terms of the sequence do not approach a specific value as n approaches infinity. For example, the sequence sin(n) does not have a limit because the sine function oscillates between -1 and 1 and does not approach a specific value as n increases.

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