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 value the function approaches as x goes to positive or negative infinity.

A point where the function is undefined and approaches infinity or negative infinity.

Look for a line that the function approaches as x goes to positive or negative infinity, but is not horizontal.

It approaches positive infinity as x approaches 0 from the right and negative infinity as x approaches 0 from the left.

It approaches positive infinity as x approaches 0 from both the left and right.

It means the function's value equals the value of the horizontal asymptote at that specific x-value, but it still approaches the asymptote as x goes to infinity.

A hole appears as an open circle on the graph at a specific x-value where the function is undefined but doesn't have a vertical asymptote.

It suggests that the function either approaches infinity or negative infinity, or has a slant asymptote.

Observe the y-value that the graph approaches as x moves further and further to the right or left.

It typically indicates a more complex function or a trigonometric component, rather than a simple rational function.

$f(x) = \frac{P(x)}{Q(x)}$, where P(x) and Q(x) are polynomials.

If $f(x) = \frac{ax^n + ...}{bx^n + ...}$, the horizontal asymptote is $y = \frac{a}{b}$.

If the degree of the denominator is greater, the horizontal asymptote is $y = 0$.

$\lim_{x \to \pm \infty} f(x) = b$

Perform polynomial long division. The quotient (without the remainder) is the slant asymptote.

Solve for $x$ in the equation $Q(x) = 0$, where $Q(x)$ is the denominator of the rational function, after simplifying the fraction.

The function tends to $\infty$ or $-\infty$, or has a slant asymptote.

$y = mx + b$, where $m$ and $b$ are constants.

Consider the signs of the leading coefficients of the numerator and denominator when $x$ is very large (positive or negative).

$f(x) = \frac{a(x-r_1)(x-r_2)...}{b(x-s_1)(x-s_2)...}$, where $r_i$ are roots of the numerator and $s_i$ are roots of the denominator.