To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching a certain limiting value "k". The precise mathematical formulation was given by Cauchy in the 19th century and is based on the notion of limit. Suppose that the graph does not have a break or a sharp edge at "p" and it is neither plumb nor too wiggly near "p". Then there is a unique value of "k" such that, as "h" approaches 0, the difference quotient gets closer and closer to "k", and the distance between them becomes negligible compared with the size of "h", if "h" is small enough. This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function "f". This limit is the derivative of the function "f" at "x" = "a", denoted "f" ′("a"). Using derivatives, the equation of the tangent line can be stated as follows:

Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the power function, trigonometric functions, exponential function, logarithm, and their various combinations. Thus, equations of the tangents to graphs of all these functions, as well as many others, can be found by the methods of calculus.

Tangent circles
More rigorous description
Analytical approach
Tangent line to a curve