Question

How do you find all the asymptotes for `R(x)=(3x+5) / (x-6)`?

Answer 1

Horizontal: `y=3`, for `x\to \pm \infty`.
Vertical: `\pm \infty`, for `x \to 6^\pm`.

Explanation:

The function is defined where the denominator is not zero, which means that all points are ok except for `x=6`. So, the domain is `(-infty,6) cup (6,infty)`.

The possible asymptotes are thus found computing the limits of `R(x)` as `x \to -\infty`, `x\to 6^{-}`, `x \to 6^{+}`, `x \to infty`.

• `lim_{x \to -\infty} R(x) = lim_{x \to -\infty} (x(3+5/x))/(x(1-6/x))`
  You can simplify the `x`'s, and both `5/x` and `-6/x` tend to zero. So, the limit is `3`.

• For the limits in `6`, from left and right, we have that the denominator tends to zero, so the function will tend to positive or negative infinity, depending on the signs: if `x\to 6^{-}`, then the denominator tends to zero from negative values, and the numerator is positive, so the limit is `-infty`. On the contrary, if `x\to 6^{+}`, both numerator and denominator are positive, so the limit is `infty`.

• The limit for `x\to\infty` requires exactly the same calculations as the one for `x\to-\infty`, so the asymptote will be `y=3` again.

• Since the function has horizontal asymptotes, there can't be oblique ones.