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This category contains results about Tangents.
Definitions specific to this category can be found in Definitions/Tangents.

Let $f: \R \to \R$ be a real function.

Let the graph of $f$ be depicted on a Cartesian plane.


Let $A = \tuple {x, \map f x}$ be a point on $G$.

The tangent to $f$ at $A$ is defined as:

$\ds \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$

Thus the tangent to $f$ at $x$ can be considered as the secant $AB$ to $G$ where:

$B = \tuple {x + h, \map f {x + h} }$

as $B$ gets closer and closer to $A$.

By taking $h$ smaller and smaller, the secant approaches more and more closely the tangent to $G$ at $A$.

Hence the tangent to $f$ is a straight line which intersects the graph of $f$ locally at a single point.

Also see