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