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# Asymptotes

An asymptote is a straight line tangent to a function at infinity.

• A vertical asymptote is a line of the form $x=a$ when $$\lim_{x\to \pm a}f(x) = \pm\infty.$$ The function has an infinite limit at $a$ when $x$ approaches $a$ either from below ($x\to a-$) or from above ($x\to a+$).
• A horizontal asymptote is a line of the form $y=b$ when $$\lim_{x\to \pm\infty}f(x) = b.$$ The function has a limit when $x$ goes to infinity.
• An oblique asymptote is a line $y = ax + b$ ($a\ne 0$) when $$\lim_{x\to\pm\infty}\big(f(x)-ax\big) = b.$$

$f(x) = 2(x-1)/x$ has two asymptotes: $x = 0$ and $y=2$.

Asymptotes: Vertical $(x=0)$ horizontal $(y=1)$ and oblique $(y=x-1)$