Asymptote

Asymptotes occur when denominator = 0, and it doesn’t cancel. x/ x+2. asymptote at x = -2

Cancelling however, does not have asymptotes, but they have Hole BUT, x/x. = 1, it has a hole at (0,1)

Vertical Asymptote

Let the denominator equal zero. This would be when:

• ${\lim }_{x\to a}=\infty$
• ${\lim }_{x\to a}=-\infty$
• ${\lim }_{x\to a^{-}}=\infty$
• ${\lim }_{x\to a^{-}}=-\infty$
• ${\lim }_{x\to a^{+}}=\infty$
• ${\lim }_{x\to a^{+}}=-\infty$

Horizontal Asymptote

Horizontal asymptotes occur at the extremes. This would be:

• ${\lim }_{x\to \infty }=L$
• ${\lim }_{x\to -\infty }=M$

3 Trials

You can run 3 tests to find the horizontal asymptote. with a rational function in standard form: $\frac{a{x}^m+\dots }{b{x}^n+\dots }$ if m < n, y = 0 if m > n, NO HA. You have a oblique or parabolic depending on the difference in degree if m = n, y = a/b

Chart Method

Make a Behavior Table Method chart Example with 1/2x-3

x	y
3/2^+	inf
3/2^-	-inf
+inf	0^+
-inf	0^-
Vertical asymptote: x = 3/2
Horizontal asymptote: y = 0
X intercept: none
Y intercept: -1/3
domain: x ∈ ℝ, x != 3/2
range: y ∈ ℝ, y ! = 0

Shit acronym

BOBO BOTN EATS D/C Bigger On the Bottom: 0 EATS D/C means “Exponents are the same divide coefficients”

Slant Asymptote

Let $m,b\in R$. The line $y=mx+b$ is a slant asymptote to the graph if either:

• ${\lim }_{x\to \infty }[f(x)-(mx+b)]=0$
• ${\lim }_{x\to -\infty }[f(x)-(mx+b)]=0$

Finding Slant Asymptote

Example: Find the slant asymptote of $\pi x+7+e^{\frac{1}{x^2}}$ Note that ${\lim }_{x\to \pm \infty }{e}^{\frac{1}{x^2}}=1$ Then consider $\pi x+7+1=\pi x+8$ Consider ${\lim }_{x\to \pm \infty }[f(x)-(\pi x+8)]$ ${\lim }_{x\to \pm \infty }[\pi x+7+e^{\frac{1}{x^2}}-(\pi x+8)]$ $={\lim }_{x\to \pm \infty }[-1+e^{\frac{1}{x^2}}]$ $={\lim }_{x\to \pm \infty }[-1]+{\lim }_{x\to \pm \infty }[e^{\frac{1}{x^2}}]$ $=-1+1$ $=0$