# Thread: limit of a function with positive integer

1. ## limit of a function with positive integer

hello,i tried to calculate the following limit but i couldn't get any result, i tried using the derivative of the numerator function but couldn't

any help is highly appreciated.

2. ## Re: limit of a function with positive integer

\displaystyle \begin{align*} \left( \frac{\sqrt{\frac{\sin{(2nx)}}{1+\cos{(nx)}}}}{(x-\frac{\pi}{2n})(x+\frac{\pi}{2n})4n^2}\right)^2 &= \frac{\frac{\sin{(\pi + 2nx - \pi)}}{1+\cos{(nx)}}}{(2nx-\pi)^2(2nx+\pi)^2} \\ &= \frac{\frac{\sin{(\pi)}\cos{(2nx - \pi)} + \cos{(\pi)}\sin{(2nx - \pi)}}{1+\cos{(nx)}}}{(2nx-\pi)^2(2nx+\pi)^2} \\ &= \frac{\frac{-\sin{(2nx - \pi)}}{1+\cos{(nx)}}}{(2nx-\pi)^2(2nx+\pi)^2} \\ &= - \frac{1}{2nx-\pi} \cdot \frac{\sin{(2nx-\pi)}}{2nx-\pi} \cdot \frac{1}{1+\cos{(nx)}} \cdot \frac{1}{(2n+x)^2} \end{align*}

So I think we are heading to $-\infty$.

(There's an error in here somewhere, I think, but my conclusion remains).

4. ## Re: limit of a function with positive integer

How can a square have a limit of negative infinity?

5. ## Re: limit of a function with positive integer

Originally Posted by HallsofIvy
How can a square have a limit of negative infinity?
$\displaystyle \lim_{x\to \infty}~(i x)^2 = -\infty$

but as the expression in the OP is real I'm just being a wiseguy and Halls is correct.

6. ## Re: limit of a function with positive integer

Originally Posted by romsek
$\displaystyle \lim_{x\to \infty}~(i x)^2 = -\infty$

but as the expression in the OP is real I'm just being a wiseguy and Halls is correct.
Wolframalpha agrees that the limit is negative infinity when approaching from the left. It is negative $i$ times infinity when approaching from the right (assuming $n$ is positive).

7. ## Re: limit of a function with positive integer

Originally Posted by SlipEternal
Wolframalpha agrees that the limit is negative infinity when approaching from the left. It is negative $i$ times infinity when approaching from the right (assuming $n$ is positive).
Assuming you are referring to your link below it looks like Wolfram agrees that the limit of the original expression is what you state above. Not the limit of the square of it as Hall notes.

8. ## Re: limit of a function with positive integer

Originally Posted by romsek
Assuming you are referring to your link below it looks like Wolfram agrees that the limit of the original expression is what you state above. Not the limit of the square of it as Hall notes.
Approaching from the right (with $n$ positive), it says that the square would approach negative infinity. The limit takes the square root of negative sign(n). Not sure how it came to that conclusion, though. I thought it went to positive infinity, but I did not feel like looking into it.

9. ## Re: limit of a function with positive integer

one thing you have to be careful with when using WA or Mathematica is that it makes no assumptions on the types of numbers being dealt with.

In particular here it assumes both $x$ and $n$ are complex numbers when I believe it's meant that $x$ is real, and $n$ is at least an integer if not an non-negative one.

There are ways in Mathematica to tell it what types of numbers are being dealt with. WA may have this functionality as well, I don't know.

10. ## Re: limit of a function with positive integer

Originally Posted by romsek
one thing you have to be careful with when using WA or Mathematica is that it makes no assumptions on the types of numbers being dealt with.

In particular here it assumes both $x$ and $n$ are complex numbers when I believe it's meant that $x$ is real, and $n$ is at least an integer if not an non-negative one.

There are ways in Mathematica to tell it what types of numbers are being dealt with. WA may have this functionality as well, I don't know.
How would $x$ approach from the left or the right if it is a complex value?

11. ## Re: limit of a function with positive integer

Originally Posted by HallsofIvy
How can a square have a limit of negative infinity?
As I said, there's an error somewhere, but the key points of the method should be the same and the conclusion is that the original function heads to negative infinity from below.