# When and how do rational functions goes to infinity

I was reading this definition of the index of a rational function.

Let $$f(x) = \frac{P(x)}{Q(x)}$$ where $$P(x)$$ is a polynomial of degree $$n$$ and $$Q(x)$$ a polynomial of degree $$n + 1$$

$$K$$ is the number of times $$f(x)$$ goes from $$+\infty$$ to $$-\infty$$

$$K'$$ is the number of times $$f(x)$$ goes from $$-\infty$$ to $$+\infty$$

The index of $$f(x)%$$ is defined as $$K - K'$$

Question: is it possible for $$f(x)$$ to go from $$+\infty$$ to $$+\infty$$? like the plot below

Edit: I think it's possible. For example $$\frac{x+1}{x^2}$$. Then the definition of index seems problematic.

The reason I am asking about this is that I am trying to solve this problem 12* which uses the definition of index.

• Are you asking if there exists an $a$ such that $Q(a)=0$ and $\lim_{x\to a}f(x)=\infty$, from the left and right of $a$? Commented Oct 13, 2021 at 1:39
• Yeah, take $Q(x) = x^2P(x)$ Commented Oct 13, 2021 at 1:41
• Please don't ask multiple questions in one post. Please do not link to a picture or outside site for the complete information about the question you are asking. Commented Oct 13, 2021 at 2:04
• You said "I am also trying to solve..." That suggests a separate question, rather than "The reason I am asking about this that I am trying to solve..." In any case, asking people to go to a separate site, so they can try to puzzle out what you mean from a "book view" seems more like imposing on people's patience than trying to help them understand where you are coming from. Commented Oct 13, 2021 at 2:27
• It is not a problematic definition. Rational functions having a pole, at which the function "goes" from $+\infty$ to $+\infty$ definitely exist, but such poles do not contribute to this definition of "index" (nowadays, I think the meaning of "index" changed). Problem 12 seems interesting too by the way. One way to approach it may be using partial fraction expansion. Another may be using contour integrals. Commented Oct 13, 2021 at 2:52