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.

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.