Chapter 3 | Derivatives
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Figure 3.5 For values of x close to 1, the graph of f ( x ) = x and its tangent line appear to coincide.
Formally we may define the tangent line to the graph of a function as follows.
Definition Let f ( x ) be a function defined in an open interval containing a . The tangent line to f ( x ) at a is the line passing through the point ⎛ ⎝ a , f ( a ) ⎞ ⎠ having slope (3.3) m tan = lim x → a f ( x )− f ( a ) x − a provided this limit exists. Equivalently, we may define the tangent line to f ( x ) at a to be the line passing through the point ⎛ ⎝ a , f ( a ) ⎞ ⎠ having slope (3.4) m tan = lim h →0 f ( a + h )− f ( a ) h provided this limit exists. Just as we have used two different expressions to define the slope of a secant line, we use two different forms to define the slope of the tangent line. In this text we use both forms of the definition. As before, the choice of definition will depend on the setting. Now that we have formally defined a tangent line to a function at a point, we can use this definition to find equations of tangent lines.
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