limit of successions, limit of n to infty of x(n)*y(n) = -3, lim y(n) = 0

[Mattiatore]

Given that {x(n)} and {y(n)} are two successions and limit of n to infty of x(n)*y(n) = -3 and limit of n to infty of y(n) = 0 then what is necesseraly true?
1) abs(x(n)) tends to infty or 2) x(n) tends to negative infty 3) x(n) tends to positive infty . I was thinking the the first one was right because y(n) could be approaching zero with negative values or postive ones, therefore we just know that x(n) goes to infty not the sign...

 

[ksdhart2]

First off, I'm guessing you made a minor typo here and the second answer is meant to say "...tends to positive infinity." If that's not the case, please explain the difference between "negative + infinity" and "negative - infinity."

For now, I'll assume it was a typo. So let's think about what we have here. I'll use "the limit of" as shorthand for "the limit as n approaches infinity of..." We know that the limit of x(n) y(n) is -3 and the limit of y(n) is 0. 0 times any real number is 0, but if x(n) approaches either positive or negative infinity, then the limit has indeterminate form and could end up being any real number. One pair of functions that satisfy all of the given criteria is y(n) = 1/x and x(n) = -3x. Here, x(n) approaches negative infinity, y(n) approaches 0, and x(n) y(n) approaches -3. Can you think of any other pair of functions where x(n) approaches positive infinity but x(n) y(n) still approaches -3? If you can find even one pair of functions that satisfy the criteria where x(n) approaches positive infinity, you'll have proven that neither 2) nor 3) can be the correct answer.

 

[Mattiatore]

Hi! Thank you very much for your answer and yes you are right it was a typo... I was thinking at x(n) = -1/n and y(n) = 3n and it should work!

 

[ksdhart2]

That seems reasonable to me, if you switch x(n) and y(n). Here, you have y(n) going to positive infinity, when the problem specifies it's meant to go to 0. But with that minor change, evaluating the limit of x(n) y(n) indeed gives -3, and x(n) goes to positive infinity. Well done.

Edit: Missed a negative sign. Whoops!

 

[Mattiatore]

Thanks for helping!