prove best lin. approx. of f(x) near 'a' is f(a+h)=f(a)+....

[jccxyz]

For the best linear approximation of f(x) near a my text says f(a+h) = f(a) + f'(a)h aproximately.

How do I prove this using the difference quotient or the derivative defination ( or both).

 

[pka]

By definition of derivative $\displaystyle \lim _{h \to 0} \frac{{f(a + h) - f(a)}}{h} = f'(a)$.
This means that if $\displaystyle h \approx 0$, h is near zero, then $\displaystyle \frac{{f(a + h) - f(a)}}{h} \approx f'(a)$.
Rearrange that to get: $\displaystyle f(a + h) \approx f(a) + hf'(a)$.