Horizontal: Function approaches a constant value as x goes to infinity. | Slant: Function approaches a line with a non-zero slope as x goes to infinity.

Vertical Asymptotes: Occur when the denominator is zero and the factor doesn't cancel. | Holes: Occur when a factor cancels from both numerator and denominator.

Numerator > Denominator: No horizontal asymptote, may have slant asymptote or approaches infinity. | Numerator < Denominator: Horizontal asymptote at y=0.

$f(x)$: Approaches 0 from above and below. | $g(x)$: Approaches 0 from above only.

Degrees equal: Divide leading coefficients. | Denominator higher: Horizontal asymptote is y=0.

Polynomial Long Division: Works for any divisor. | Synthetic Division: Only works for divisors of the form (x - a).

y=2: The function approaches the line y=2 as x approaches infinity. | y=0: The function approaches the x-axis as x approaches infinity.

$f(x)$: Horizontal asymptote at y=1, vertical asymptote at x=1. | $g(x)$: Horizontal asymptote at y=1, vertical asymptotes at x=1 and x=-1.

Even powers: Function approaches the horizontal asymptote from the same side for both positive and negative infinity. | Odd powers: Function approaches the horizontal asymptote from opposite sides for positive and negative infinity.

Vertical asymptotes: Exclude a value from the domain where the function is undefined and approaches infinity. | Holes: Exclude a value from the domain where the function is undefined, but the limit exists.

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.

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 $x=-2$.