Limit of n/(n+1)-n/(n+2) as n->Infinity: 0 or 1?

In summary: However, when taking the limit as n approaches infinity, we must be careful with the order of operations. In the first expression, n/(n+1)-n/(n+2), we can simplify to (n^2-n)/(n^2+2n). When taking the limit, the n^2 terms become dominant, resulting in a limit of 0. In the second expression, n/(n+1)/(n+2), we can simplify to (n/(n+1))/(n+2). When taking the limit, the (n+1) term becomes dominant, resulting in a limit of 1. This highlights the importance of using proper order of operations when dealing with limits. In summary, the expressions are
  • #1
pivoxa15
2,255
1
Take n/(n+1)-n/(n+2) as n->infinity results in the limit 0

But take the same equation convert it to n/(n+1)/(n+2) as n-> infinity results in the limit 1.

What is going on? The expression is the same yet you get two different limits. Something is wrong.
 
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  • #2
pivoxa15 said:
Take n/(n+1)-n/(n+2) as n->infinity results in the limit 0

Correct.

But take the same equation convert it to n/(n+1)/(n+2) as n-> infinity results in the limit 1.

How? The expression is [n/(n+1)]/(n+2). Working from left to right, the limit reduces to (n/n)/(n+2) = 1/(n+2) = 0. (Very sloppy, but I can't do the tex now).
 
  • #3
I see. I made a mistake in my OP.
 
  • #4
pivoxa15 said:
Take n/(n+1)-n/(n+2) as n->infinity results in the limit 0

But take the same equation convert it to n/(n+1)/(n+2) as n-> infinity results in the limit 1.

What is going on? The expression is the same yet you get two different limits. Something is wrong.
?? And if you were to "convert" it into something else it is not equal to you would get yet a different answer! Garbage in, garbage out.
 
  • #5
HallsofIvy said:
?? And if you were to "convert" it into something else it is not equal to you would get yet a different answer! Garbage in, garbage out.

The expressions are identically equal.
 

FAQ: Limit of n/(n+1)-n/(n+2) as n->Infinity: 0 or 1?

What is the limit of n/(n+1)-n/(n+2) as n->Infinity?

The limit of n/(n+1)-n/(n+2) as n->Infinity is 0.

How do you find the limit of a function as n approaches Infinity?

To find the limit of a function as n approaches Infinity, you can plug in larger and larger values for n and see if the function approaches a specific value. Alternatively, you can use algebraic techniques such as factoring or rationalizing the denominator to simplify the function and determine the limit.

Can the limit of a function as n->Infinity be a negative number?

Yes, the limit of a function as n->Infinity can be a negative number. It depends on the behavior of the function as n gets larger and larger.

What does it mean if the limit of a function as n->Infinity is undefined?

If the limit of a function as n->Infinity is undefined, it means that the function either approaches infinity or negative infinity as n gets larger and larger. This can also occur if there is a vertical asymptote in the function.

Can the limit of a function as n->Infinity be a fraction?

Yes, the limit of a function as n->Infinity can be a fraction. It depends on the behavior of the function as n gets larger and larger.

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