Question

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

Answer 1

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^+)`