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The degree of the polynomial is the highest exponent occurring in the polynomial.

Leading coefficients are the numbers written in front of the variable with the largest exponent.

For instance, $P(x) = x^3-x^2+2x+4$ is already in descending order.

However, $Q(x) = x-x^4+1$ is not in the correct form. We may rewrite it as $Q(x) = -x^4+x+1$.

The degree of the polynomial is the highest exponent occurring in the polynomial.

For instance, the degree of $P(x) = x^3-x^2+2x+4$ is $p=3$, and that of $Q(x) = -x^4 +x +1$ is $q=4$.

Leading coefficients are the numbers written in front of the variable with the largest exponent.

For instance, the leading coefficients of $P(x) = x^3-x^2+2x+4$ is $a=1$, and that of $Q(x) = -x^4 +x +1$ is $b=-1$.

We can have three possibilities:

1. The degree of $P(x)$ is higher.

2. The degree of $Q(x)$ is higher.

3. The degrees are equal.

We determine the leading coefficients of $P(x)$ and $Q(x)$ (the coefficient of the term with the highest exponent).

If the leading coefficients are of the same sign, then $$\lim_{x\to \infty}{ R(x)} = \infty$$ and $$\lim_{x\to -\infty}{ R(x)} = -\infty.$$

If they are of opposite signs, then $$\lim_{x\to \infty}{ R(x)} = -\infty$$ and $$\lim_{x\to -\infty}{ R(x)} = \infty.$$

This case is the simplest!

We have $$\lim_{x\to \infty}{ R(x)} = \lim_{x\to -\infty}{ R(x)} = 0.$$

The answers depend on the signs of the leading coefficients.

We assume $P(x) = ax^p +\cdots$ and $Q(x) = bx^p +\cdots$, so that the first coefficients are $a$ and $b$ for the two polynomials, respectively.

We have $$\lim_{x\to \infty}{ R(x)} = \lim_{x\to- \infty}{ R(x)} = \frac{a}{b}.$$

In the following, we give several examples to illustrate how we can apply the above rules to the different cases.

Consider $$R(x) = \frac{4x^3 + 5x +4}{- x^2 +2x +1}.$$

The degree of $P(x) = 4x^3 + 5x +4$ is $3$ and that of $Q(x) = -x^2 +2x +1$ is $2$. So this example belongs to case 1 ($P(x)$ has the higher degree). Furthermore, the leading coefficient of $P(x)$ is positive, while that of $Q(x)$ is negative, so they are of opposite signs.

Therefore, we have $$\lim_{x\to \infty}{ R(x)} = -\infty$$ and $$\lim_{x\to -\infty}{ R(x)} = \infty.$$

Let's return to our primary example: $P(x) = x^3-x^2+2x+4$, and $Q(x) = -x^4 +x +1$.

The rational function in question is $$R(x) = \frac{x^3-x^2+2x+4}{ -x^4 +x +1}.$$So this corresponds to the second case ($Q(x)$ has a higher degree), and we know

$$\lim_{x\to \infty}{ R(x)} = \lim_{x\to -\infty}{ R(x)} = 0.$$

Consider $$R(x) = \frac{4x^2 + 2x +3}{ -3x^2 +2x +1}.$$

The degree of $P(x) = 4x^2 + 2x +3$ is $2$ and that of $Q(x) = -3x^2 +2x +1$ is $2$.

So this example belongs to case 3, where $P(x)$ and $Q(x)$ have equal degrees.

The leading coefficient of $P(x)$ is $a=4$, while that of $Q(x)$ is $b=-3$, so we have $$\lim_{x\to \infty}{ R(x)} = \lim_{x\to- \infty}{ R(x)} = \frac{a}{b}= \frac{4}{-3} = -\frac{4}{3}.$$