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 $x=-2$.

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.

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.