List of formulas and rules related to the asymptotes of the graph of a function:
- Vertical asymptotes:
- A vertical asymptote occurs when a function tends to infinity as it approaches a certain value ( x = a ). This occurs if:
[
\lim_{x \to a} f(x) = \pm \infty
] - Typically, vertical asymptotes are found at points where the function is undefined (for example, division by zero).
- Horizontal asymptotes:
- The horizontal asymptote describes the behavior of a function as ( x ) tends to infinity. If:
[
\lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L
]
then ( y = L ) is a horizontal asymptote.
- Oblique (or slanted) asymptotes:
- A slant asymptote exists if:
[
\lim_{x \to \infty} \left( f(x) – (mx + b) \right) = 0
]
where ( m ) and ( b ) are the coefficients that determine the slope and intercept with the ( y ) axis. Typically, slant asymptotes occur when the degree of the numerator of a function is greater than the degree of the denominator by 1.
- General rules:
- For rational functions ( \frac{P(x)}{Q(x)} ):
- If the degree ( P < ) is the degree ( Q ), then the horizontal asymptote is ( y = 0 ).
- If the degree ( P = ) is the degree ( Q ), then the horizontal asymptote is ( y = \frac{a}{b} ), where ( a ) and ( b ) are the leading coefficients of ( P ) and ( Q ).
- If the degree ( P > ) is the degree ( Q ), then the oblique asymptote can be found by dividing the polynomials.