# 3.9 | Linear Approximation and the Derivative

Suppose $$f$$ is differentiable at $$a$$. Then, for values of $$x$$ near $$a$$, the tangent line approximation of $$f(x)$$ is
$$f(x) \approx f(a) + f'(a)(x-a).$$
The expression $$f(a) + f'(a)(x-a)$$ is called the local linearization of $$f$$ near $$x = a$$. We are thinking of $$a$$ as fixed, so that $$f(a)$$ and $$f'(a)$$ are constant.
The error, $$E(x)$$, in the approximation is defined by
$$E(x) = f(x) – f(a) – f'(a)(x-a).$$
Suppose $$f$$ is differentiable at $$x = a$$ and $$E(x)$$ is the error in the tangent line approximation, that is:
$$E(x) = f(x) – f(a) -f'(a)(x-a).$$
Then
$$\lim_{x \rightarrow a} \frac{E(x)}{x-a} = 0.$$