What is the limit of (n/n+1) as n approaches infinity?

[Firepanda]

I believe the second is simply 1, as I can ignore a here.

Not sure about the first, I believe it tends to 0 because of the power of n, and (n/n+1) < 1

Any help appreciated, thanks!

 

[TheFurryGoat]

For the first:
First of all, you need to fiddle around with the expression so that you get something you recognize. Now

n/(n+1) = 1 / ( (n+1)/n ) = 1 / ( (1+1/n) ),

so if you plug this in the original you get

( 1 / ( (1+1/n) ) )^n = 1 /( ( (1+1/n) ) )^n

and now you should recognize the limit of this.

 

[Firepanda]

For the first:
First of all, you need to fiddle around with the expression so that you get something you recognize. Now

n/(n+1) = 1 / ( (n+1)/n ) = 1 / ( (1+1/n) ),

so if you plug this in the original you get

( 1 / ( (1+1/n) ) )^n = 1 /( ( (1+1/n) ) )^n

and now you should recognize the limit of this.

Looks like 0 to me, but only because 1+1/n > 1 and the power of n

Which is the same reasoning as I had before.. so I'm still lost!

 

[SammyS]

In the Calculus & Beyond section, you should recognize:

$\displaystyle \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\,.$​

If not, try increasingly larger numbers for n in your calculator: 1 , 2 , 3 , 5 , 10 , 1000 , 1000000 , ...

 

[TheFurryGoat]

Have you encountered the limit of
(1 + 1/n)^n
before? I'm sure it has been mentioned somewhere in your study material. The limit is Euler's number $e\ =\ 2.7182...$.

 

[Mark44]

The first limit is an example of the indeterminate form [1]^∞.