Question

Determine if, lim as h goes to 0 of (f(1+h)-f(1))/h exists when f(x)=x^2-4x and if it does find its value

Answer 1

\[f(x)=x^2-4x=>f(1+h)=(1+h)^2-4(1+h) \& f(1)=(1)^2-4(1)\]

Answer 2

\[f(1+h)=1^2+2\cdot 1 \cdot h+h^2-4-4h=1+2h+h^2-4-4h=h^2-2h-3\]
\[f(1)=1-4=-3\]

Answer 3

\[\lim_{h \rightarrow 0}\frac{f(1+h)-f(1)}{h}=\lim_{h \rightarrow 0}\frac{[h^2-2h-3]-[-3]}{h}\]

Answer 4

\[\lim_{h \rightarrow 0}\frac{h^2-2h}{h}=\lim_{h \rightarrow 0}\frac{h(h-2)}{h}=\lim_{h \rightarrow 0}(h-2)=0-2=-2\]