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What does it mean about a point on a function if the tangent line crosses the function at that point?

Answer 1

I'd say it means either the tangent line is vertical of the point of tangency is an inflection point. BUT

Without a precise definition of what it means to say "a tangent line crosses the function at that point", I'm not sure what the question really means.

The tangent line to the graph of #f(x) = tanx# at #(0,0)# seems to "cross the graph of the function" at #(0,0)# I mention this example because it makes it clear that the tangent line need not be horizontal.

In the absense of a definition it is not clear whether the tangent to the graph of #g(x) = x^(2/3)# at #(0,0)# "crosses the graph". The tangent at that point is vertical, and #(0,0)# is not an inflection point.

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Answer 2

William Abdalla

If the tangent line crosses the function at a point, it means that the function is differentiable at that point. The tangent line represents the instantaneous rate of change of the function at that specific point.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.