Question

Find


`\lim_(h \to \0) ( f(4+h)-f(4))/(h)` if f(x) = absolute value of (4x-1)

Cheers! :)

Answer 1

By definition of the absolute value, you have


`|4x-1|=\begin{cases}4x-1&\text{for }x\ge\frac14\\1-4x&\text{for }x<\frac14\end{cases}`

For small
`h`, you have
`4+h\approx4`, so you would write
`|4x-1|=4x-1`. The limit is then


`\displaystyle\lim_(h\to0)\frac{f(4+h)-f(4)}h=\lim_(h\to0)\frac{(4(4+h)-1)-(4(4)-1)}h=\lim_(h\to0)\frac{4h}h=\lim_(h\to0)4=4`