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 $x=-2$.

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.

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.