For one thing, you have . With respect to , this is a constant function (and therefore has no critical points). I'm not sure if that's what you meant, but with the case , we have:
Therefore,
Using the quadratic formula on the numerator, we get that has two critical points (with horizontal tangent lines) at:
is defined everywhere, so there are no vertical tangents.
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On a related note, it only makes sense to define a vertical tangent of at when:
1) is undefined.
2) is defined.
Otherwise, you'll just get the equation for the asymptote.
Take for example, and investigate the behavior of the tangent line at .