Is y = -1/∞ an asymptote in y = x^2 ?

1 Answer
George C.
Jun 15, 2017

#y=x^2# has no (linear) asymptotes

Explanation:

An linear asymptote is a line towards which a curve approaches arbitrarily close.

More precisely, a function #f(x)# can have three kinds of asymptote:

  • A horizontal asymptote is a horizontal line towards which #f(x)# tends as #x->oo# or as #x->-oo# (or both).

  • A vertical asymptote is a vertical line #x=a# such that #lim_(x->a^-) f(x) = +-oo# and #lim_(x->a^+) f(x) = +-oo#.

  • An oblique (a.k.a. slant) asymptote is any other line towards which #f(x)# tends as #x->oo# or #x->-oo# or both.

If the expression has any meaning, then #-1/oo = 0#, so you are asking whether the horizontal line #y=0# is a horizontal asymptote of #f(x) = x^2#. No, since #f(x)->oo# as #x->+-oo#, it does not tend towards this line.

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