Answer:
I can't see the options, so i will find all the asymptotes.
We have the function:
![f(x) = \frac{x^4 - 4*x^2 + x}{-2*x^4 + 18*x^2}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7Bx%5E4%20-%204%2Ax%5E2%20%2B%20x%7D%7B-2%2Ax%5E4%20%2B%2018%2Ax%5E2%7D)
First, we can graph this using a graphing tool, the graph can be seen below.
In the graph, we can see that when we approach x = 0 from the left, f(x) goes to negative infinity, while if we approach x = 0 from the right, f(x) goes to infinity.
This can be written as:
![\lim_{x \to 0_-} f(x) = - \infty \\](https://tex.z-dn.net/?f=%5Clim_%7Bx%20%5Cto%200_-%7D%20f%28x%29%20%3D%20-%20%5Cinfty%20%5C%5C)
and:
![\lim_{x \to 0_+} f(x) = + \infty \\](https://tex.z-dn.net/?f=%5Clim_%7Bx%20%5Cto%200_%2B%7D%20f%28x%29%20%3D%20%2B%20%5Cinfty%20%5C%5C)
A similar thing can be seen at x = 3, when we approach from the left f(x) goes to infinity, while if we approach from the left, f(x) goes to negative infinity.
Then:
![\lim_{x \to 3_-} f(x) = \infty \\\\\lim_{x \to 3_+} f(x) = - \infty \\](https://tex.z-dn.net/?f=%5Clim_%7Bx%20%5Cto%203_-%7D%20f%28x%29%20%3D%20%20%5Cinfty%20%5C%5C%5C%5C%5Clim_%7Bx%20%5Cto%203_%2B%7D%20f%28x%29%20%3D%20-%20%5Cinfty%20%5C%5C)
For x = -3 we can see that when we approach from the left, f(x) goes to negative infinity, while if we approach from the right, f(x) goes to infinity.
Then:
![\lim_{x \to -3_-} f(x) = - \infty \\\\\lim_{x \to -3_+} f(x) = + \infty \\](https://tex.z-dn.net/?f=%5Clim_%7Bx%20%5Cto%20-3_-%7D%20f%28x%29%20%3D%20-%20%5Cinfty%20%5C%5C%5C%5C%5Clim_%7Bx%20%5Cto%20-3_%2B%7D%20f%28x%29%20%3D%20%2B%20%5Cinfty%20%5C%5C)
We also can see that as x goes to negative infinity or positive infinity, f(x) tends to -0.5
Then:
![\lim_{x \to \infty} f(x) = -0.5 \\ \lim_{x \to -\infty} f(x) = -0.5](https://tex.z-dn.net/?f=%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20f%28x%29%20%3D%20-0.5%20%5C%5C%20%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%20f%28x%29%20%3D%20-0.5)
So you need to check the options that match with some of the given tendencies.