Question

Find the slope of the function -2x + 1 by the definition of limit. a0

Answer 1

Let $f(x)=-2x+1$. By definition, the derivative is

$f'(x)=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}h$

So

$f'(x)=\displaystyle\lim_{h\to0}\frac{(-2(x+h)+1)-(-2x+1)}h=\lim_{h\to0}\frac{-2h}h=\lim_{h\to0}-2=-2$

and the slope of any line tangent to $f(x)$ is -2.