limΔ x→ 0[((x+Δ x)²-2(x+Δ x)+1-(x²-2x+1))/(Δ x)]

7.2 Q.II · Q29

\displaystyle\lim_{\Delta x\to 0}\left[\frac{(x+\Delta x)^2-2(x+\Delta x)+1-(x^2-2x+1)}{\Delta x}\right]

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Expanding: $(x+\Delta x)^2-2(x+\Delta x)+1=x^2+2x\Delta x+\Delta x^2-2x-2\Delta x+1$ . Subtracting $x^2-2x+1$ leaves $2x\Delta x+\Delta x^2-2\Delta x=\Delta x(2x+\Delta x-2)$ . Dividing by $\Delta x$ (valid since $\Delta x\to0$ but …

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Q. $\displaystyle\lim_{\Delta x\to 0}\left[\frac{(x+\Delta x)^2-2(x+\Delta x)+1-(x^2-2x+1)}{\Delta x}\right]$

Concept understanding — Limits by Factorization

Q.lim⁡Δx→0[(x+Δx)2−2(x+Δx)+1−(x2−2x+1)Δx]\displaystyle\lim_{\Delta x\to 0}\left[\frac{(x+\Delta x)^2-2(x+\Delta x)+1-(x^2-2x+1)}{\Delta x}\right]Δx→0lim​[Δx(x+Δx)2−2(x+Δx)+1−(x2−2x+1)​]

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Q. $\displaystyle\lim_{\Delta x\to 0}\left[\frac{(x+\Delta x)^2-2(x+\Delta x)+1-(x^2-2x+1)}{\Delta x}\right]$

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