Limits splitting the fraction into two

In summary, the conversation discusses a challenging mathematical problem involving the limit of (x2n-1)/(x2n+1). Various approaches are suggested, including splitting the fraction, using logarithms, factoring, and applying L'Hopital's rule. It is suggested that considering different cases, such as x>1, x=1, and x<0, may lead to a solution. It is also noted that the value of n may need to be an integer to avoid issues with fractional powers of negative numbers.
  • #1
StephenPrivitera
363
0
limn-->oo(x2n-1)/(x2n+1)
I can't figure this one out. I've tried everything. I tried splitting the fraction into two, applying a log to each side, factoring the top, dividing by x2n, and Lhopitals rule doesn't apply and wouldn't help if it did. Any ideas?
 
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  • #2
perhaps the only way to do this would be to consider various cases such as x>1, x=1, x<-1, x=-1, etc.
anyone agree?

I get for x>1
lim=1
for x=1, lim=0
if 0<x<1, lim=-1

What about x<0? What is -2^999999.5? Surely, it's not real. Can we say that lim DNE for x<0?
For -1<x<0 x^n would be very small, but wouldn't n have to be some integer?
 
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  • #3
I got stuck.

For example,
I'll omit the subscripts.

lim(x2n-1)/(x2n+1)=lim(1-1/x2n)/(1+1/x2n)
As n grows to infinity, we can say nothing about the limit, because it depends on what x is. If x is small then 1/x is large. If x is big, then 1/x is small.


edit: whoops edited wrong post, sorry.
 
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  • #4
Grargh, you read and responded before I could delete my post.

Yes, breaking it up into cases is a good idea.

edit: n often implicitly means an integer, and it wouldn't surprise me if this problem assumed as such.

*sigh* Today isn't my best day. :wink:
 
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  • #5
I would interpret this as n being an integer so you don't have any problems with fractional powers of negative numbers.

Because x2n= (x2)n) it doesn't matter whether x is positive or negative so you might as well assume positive. In that case the crucial cases are: 0<= x<1, x= 1, x> 1.
 

1. What is the purpose of splitting a fraction into two limits?

Splitting a fraction into two limits helps to simplify complex fractions and makes it easier to solve for the limit of the entire fraction. It also allows us to apply the limit properties separately to each part of the fraction.

2. Can any fraction be split into two limits?

Yes, any fraction can be split into two limits as long as the denominator is not zero. This is because the limit of a fraction only exists when the denominator is not approaching zero.

3. How do I split a fraction into two limits?

To split a fraction into two limits, first factor the numerator and denominator into separate expressions. Then, divide the fraction into two parts, with each part containing one of the expressions. Finally, take the limit of each part separately.

4. What is the rule for splitting a product of two functions into two limits?

The rule for splitting a product of two functions into two limits is to first take the limit of each function separately and then multiply the two limits together. This is known as the limit product rule.

5. Are there any limitations or exceptions to splitting a fraction into two limits?

One limitation to splitting a fraction into two limits is that it can only be applied when the limit of the entire fraction exists. Additionally, if the fraction contains trigonometric functions, logarithms, or other special functions, special rules may need to be applied when splitting the fraction into two limits.

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