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

[physics=world]

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?

 

[tiny-tim]

hi physics=world!

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 ​

 

[physics=world]

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.

 

[tiny-tim]

hi physics=world!

i don't understand this line …

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

where did the ∞ on the bottom go?

∞ is lot larger than 2 (!), so why are you ignoring it, instead of ignoring the 2 ?

 

[Ibix]

The aim with limits is to avoid writing n=\infty 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?

 

[physics=world]

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?

 

[tiny-tim]

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 ∞ !

so, the 2 is supposed to be ignored?

yup!

 

[physics=world]

so for example if it was say 5 / infinity

would the answer be zero? or infinity? or undefined?

 

[tiny-tim]

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 ∞)