Consider this function and its horizontal asymptote. Can the asymptote (in blue) also be considered a tangent line to the curve (in red)? The slope of the curve definitely approaches zero as $x$ approaches $\pm\infty$, but does that mean that a horizontal tangent line exists? It isn't possible to find a point of tangency, so I'm not sure if it counts.
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$\begingroup$The asymptotic approach is not considered a form of tangency. However, we can construct curves that are asymptotic and tangent to the same line, like the line $y=0$ with respect to the curve $y=(x^2)/(1+x^4)$.
$\endgroup$ $\begingroup$In projective geometry, if we allow a point at infinity $\{\infty\}$, then one can claim the asymptote is tangent at $\infty$.
In standard Euclidean geometry, no horizontal tangent line exists.
See for the definition and introductory discussion.
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