Question

Find the limit. (If an answer does not exist, enter DNE.) lim Δx→0 (x + Δx)2 − 3(x + Δx) + 2 − (x2 − 3x + 2) Δx

Answer 1

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

Expand the numerator as

`x^2+2x\Delta x+(\Delta x)^2-3x-3\Delta x+2-x^2+3x-2=(2x-3)\Delta x+(\Delta x)^2`

Then in the limit,

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