Simplifying an Indeterminate Quotient: Help Needed

[markcholden]

I need to take the limit of this quotient as n goes to infinity:

[2(-1)^(n+1) - 3^(n+1)]
-----------------------
[2(-1)^(n) - 3^(n)]

It seems to go to infinity over infinity in its current form, which, if I recall correctly, is indeterminate. It seems then that I need to simplify it...but I don't know how. Any ideas greatly appreciated.

 

[mathman]

The expression you have written approaches 3⁽ⁿ⁺¹⁾/3ⁿ for large n. I presume you can take it from here.

 

[Hurkyl]

Well, mathman already handed you the bit that your intuition should get you. The key there is to look at all of the pieces of the expression and figure out how they behave as n goes to infinity.

Once you've figured out that intuitive part, you already know the trick to rigorously work through that limit -- I'll leave it to you to figure out.

(By rigorously, I mean that you don't just say that the limit is obviously converging to 3ⁿ⁺¹ / 3ⁿ)

(If you really need another hint: think about rational functions, but try not to peek!)

 

[HallsofIvy]

Terms that go to infinity are hard to handle. Terms that go to 0 are easy!
It looks to me that, for any n, 3ⁿ⁺¹ will be the largest term. What happens if you divide every term in both numerator and denominator by 3ⁿ⁺¹?