by nathi123 » Thu Dec 13, 2018 1:16 pm
a) [tex]f'(2) = \lim_{x \to 2}\frac{f(x) - f(2)}{x-2}=\lim_{x \to 2}\frac{\frac{1-x}{2x}-\frac{-1}{4}}{x-2}=\lim_{x \to 2}\frac{2-x}{4x(x-2)}=\lim_{x \to 2}(-\frac{1}{4x})=-\frac{1}{8}[/tex]
b) f'(x) = [tex]\lim_{x \to x_{0 }}\frac{f(x) - f(x_{0 }}{x-x_{0 }}=\lim_{x \to x_{0 }}\frac{\frac{1-x}{2x}-\frac{1-x_{0 }}{2x_{0 }}}{x-x_{0 }}=\lim_{x \to x_{0 }}\frac{x_{0 }-x}{2xx_{0 }(x-x_{0 })}=\lim_{x \to x_{0 }}\frac{-1}{2xx_{0}}=-\frac{1}{2x_{0 }^{2}}[/tex].