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 $x=-2$.

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.

$f(x) = \frac{P(x)}{Q(x)}$, 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.