The relationship between the degrees determines whether there is a horizontal asymptote, a slant asymptote, or the function approaches infinity.

A horizontal asymptote describes the value that the function approaches as x goes to positive or negative infinity.

Vertical asymptotes occur at x-values where the denominator of the rational function equals zero and the numerator does not.

The ratio of the leading terms dominates the function's behavior as x approaches infinity, determining the horizontal or slant asymptote.

Limits provide a precise way to express what value a function approaches as x tends to infinity, indicating the function's end behavior.

A hole occurs when a factor cancels out from both the numerator and denominator, creating a point where the function is undefined but doesn't have a vertical asymptote.

Simplifying can reveal holes, and the simplified form is used to determine vertical asymptotes. End behavior and horizontal asymptotes are determined from the original function.

End behavior describes the function's trend as x approaches infinity, while local behavior describes the function's characteristics within a specific interval or around a particular point.

The sign determines whether the function approaches positive or negative infinity as x approaches positive or negative infinity.

The term with the highest degree in the numerator or denominator 'dominates' the function's behavior as x becomes very large, influencing the end behavior.

A function that is the ratio of two polynomials, expressed as $\frac{P(x)}{Q(x)}$.

How the function behaves as $x$ approaches positive or negative infinity.

A horizontal line that the graph of a function approaches as $x$ tends to $+\infty$ or $-\infty$.

An asymptote that is neither horizontal nor vertical. Occurs when the degree of the numerator is one greater than the degree of the denominator.

A vertical line $x=a$ where the function approaches infinity or negative infinity as $x$ approaches $a$.

The term with the highest power of the variable in a polynomial.

$\lim_{x \to \pm \infty} f(x) = L$, where L is the limit as x approaches infinity or negative infinity.

The highest power of the variable in the polynomial.

The result of dividing the leading term of the numerator by the leading term of the denominator in a rational function.

A fraction where the numerator and/or denominator are polynomials.

Compare degrees: Degrees are equal. Divide leading coefficients: $y = \frac{2}{1} = 2$.

Numerator degree > denominator degree. Polynomial long division gives a quotient, indicating slant asymptote or unbounded behavior.

Factor denominator: $x^2 - 4 = (x-2)(x+2)$. Set each factor to zero: $x = 2, x = -2$.

$\lim_{x \to \infty} \frac{1}{x} = 0$ and $\lim_{x \to -\infty} \frac{1}{x} = 0$.

Perform polynomial division. $\frac{x^2 + 2x + 1}{x + 1} = x+1$. Slant asymptote is $y = x + 1$.

Factor the numerator: $f(x) = \frac{(x-1)(x+1)}{x-1}$. Since (x-1) cancels, there is a hole at x=1.

Degree of numerator is higher. Divide leading terms: $\frac{5x^4}{x^2} = 5x^2$. As $x \to \pm \infty$, $f(x) \to \infty$.

Degrees are equal. Divide leading coefficients: $y=\frac{3}{1}=3$

Degree of denominator is higher. Horizontal asymptote is $y=0$.

Factor denominator: $x^2+5x+6=(x+2)(x+3)$. Simplify function: $f(x)=\frac{1}{x+2}$. Vertical asymptote at $x=-2$.