Question

Find lim ?x approaches 0 f(x+?x)-f(x)/?x where f(x) = 4x-3

Answer 1

If
`f(x)=4x-3`:

`\displaystyle\lim_(\Delta x\to0)((4(x+\Delta x)-3)-(4x-3))/(\Delta x)=\lim_(\Delta x\to0)(4\Delta x)/(\Delta x)=4`

If
`f(x)=4x^(-3)`:

`\displaystyle\lim_(\Delta x\to0)\frac{\frac4{(x+\Delta x)^3}-\frac4{x^3}}{\Delta x}=\lim_(\Delta x\to0)((4x^3-4(x+\Delta x)^3)/(x^3(x+\Delta x)^3))/(\Delta x)`

`\displaystyle=\lim_(\Delta x\to0)(4x^3-4(x^3+3x^2\Delta x+3x(\Delta x)^2+(\Delta x)^3))/(x^3\Delta x(x+\Delta x)^3)`

`\displaystyle=\lim_(\Delta x\to0)(-12x^2\Delta x-12x(\Delta x)^2-4(\Delta x)^3)/(x^3\Delta x(x+\Delta x)^3)=-(12)/(x^4)`