How can you compare the following ways of defining a tangent line to a curve at a point P(α,β) with examples?

1)A straight line meet the curve at the point P with the curve appearing in only one side of the line as x approaches α . In other words a straight line meets the curve without crossing it at P.

2) As stated in Wolfram mathworld a straight line passes through the point P on the curve with a slope of f'(α) , where f'( α) is the derivative of f(x) at x = α .