Question

Find the limit (if it exists). (If an answer does not exist, enter DNE.) $ \displaystyle{\lim_{\Delta x \rightarrow 0}} \text{ } \dfrac{{\color{red}2} (x + \Delta x) - {\color{red}2} x}{\Delta x} $

Answer 1

2x - 2

Explanation:

Here is the completer question

lim as Δx → 0 [(x + Δx)²- 2(x + Δx) + 3 - (x² - 2x + 3)]/Δx

Solution

lim as Δx → 0 [(x + Δx)²- 2(x + Δx) + 3 - (x² - 2x + 3)]/Δx

Expanding the brackets, we have

lim as Δx → 0 [(x² + 2xΔx + (Δx)²- 2x - 2Δx + 3 - x² + 2x - 3)]/Δx

Collecting like terms. we have

lim as Δx → 0 [(x² - x² + 2xΔx - 2Δx + (Δx)²- 2x + 2x + 3 - 3)]/Δx

Simplifying, we have

lim as Δx → 0 [(0 + 2xΔx - 2Δx + (Δx)² + 0 + 0)]/Δx

lim as Δx → 0 [2xΔx - 2Δx + (Δx)²]/Δx

Dividing through by Δx, we have

lim as Δx → 0 [2x - 2 + (Δx)]

Inserting Δx = 0, we have

= lim as Δx → 0 (2x -2 + 0)

= 2x -2

So lim as Δx → 0 [(x + Δx)²- 2(x + Δx) + 3 - (x² - 2x + 3)]/Δx = 2x -2