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So I understand how to graph rational functions kind of. I know how to find the vertical asymptotes, zeroes, and horizontal asymptotes, etc. What I don't understand is how to find the end behavior.

My teacher taught us that if the degree is even then the left side goes up and if its odd the left side goes down//if the coefficient is positive the right side goes up and if its negative the right side goes down.

But how do you figure all that out if it's a fraction? Do you actually have to divide the numerator by the denominator, because I'm pretty sure there's an easier way.

for example:

$\frac{-x^2(x+1)^2(x+3)}{(x-2)^2}$

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For the large $x$ behaviour, replace each factor by its dominant term (which is the one with the highest power of $x$). Thus in your example,

$$ \frac{-x^2(x+1)^2(x+3)}{(x-2)^2} \sim \frac{-x^2 \cdot x^2 \cdot x}{x^2} = - x^3 \ \text{as}\ x \to \pm \infty $$

So, like $-x^3$, this goes to $+\infty$ as $x \to -\infty$ and $-\infty$ as $x \to +\infty$.

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