How do you find the asymptotes for #R(x)= (x+2)/(x^2-64)#?

1 Answer
Dec 28, 2016

Answer:

See explanation

Explanation:

The expression becomes undefined at the point where you have:

#("some value")/0#.

So we have:

#color(blue)(lim_(x^2-64->0^+) =lim_(x->8^+) (x+2)/(x^2-64) ->+oo)#

#color(blue)(lim_(x^2-64->0^-) =lim_(x->8^-) (x+2)/(x^2-64) -> -oo)#

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Now we investigate as #x->+-oo#

As #x# becomes increasingly positive or negative the less and less influence is applied by the 2 in the numerator and the -64 in the denominator.

Thus the expression tend towards #x/x^2 = 1/(+-x)#

And as #|x| ->oo # then #1/(|x|) ->0#

#color(blue)(lim_(x->oo^-) (x+2)/(x^2-64)->0^-)#

#color(blue)(lim_(x->oo^+) (x+2)/(x^2-64)->0^+)#
Tony B