Linear approximation

• $f$ is differentiable at $x_0$

• The approximating function $L(x)=f(x_0)+f^{\prime }(x_0)(x-x_0)$ is the linearization of $f$ at $x_0$

• The approximation $f(x)\approx L(x)$ of $f$ by $L$ is the (standard) linear approximation of $f$ at $x_0$

• The point $x=x_0$ is the center of the approximation

• $f(x)\approx f(x_0)$

• $l(x)=f(x_0)+f^{\prime }(x_0)\cdot (x-x_0)$

• $r(x)=f(x)-l(x)$

• $\underset{x\to x_0}{\lim }\frac{r(x)}{x-x_0}=0$