Does the graph of f(x)={-1, x 0 . have a vertical tangent at t

step 1 Is there a vertical tangent at the origin of this function? First, let's figure out what the ori

Does the graph of f(x)={-1, x

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Key Concepts

Vertical Tangent

A vertical tangent is a line at a point on a curve where the slope becomes infinite; mathematically, this means the derivative approaches positive or negative infinity. A vertical tangent occurs when the graph of a function is locally nearly vertical, but it still requires the function to be continuous in the neighborhood of the point.

Continuity and Differentiability

For a tangent line to be defined at a point, the function must be continuous and differentiable at that point. If a function has a discontinuity—such as a jump or a hole—the concept of a tangent line, even a vertical one, does not apply because the limit defining the derivative does not exist.

Piecewise-defined Functions

Piecewise-defined functions use different expressions to define the function on different intervals. This can lead to abrupt changes in the function's behavior, including discontinuities and non-differentiable points where the different pieces meet, making it impossible to have a well-defined tangent at those junctions.

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