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Question
Evaluate the following limit :
`lim_(Deltax -> 0) [((x + Deltax)^2 - 2(x + Deltax) + 1 - (x^2 - 2x + 1))/(Deltax)]`
Solution
`lim_(Deltax -> 0) ((x + Deltax)^2 - 2(x + Deltax) + 1 - (x^2 - 2x + 1))/(Deltax)`
= `lim_(Deltax -> 0) (x^2 + 2xDeltax + (Deltax)^2 - 2x - 2Deltax + 1 - x^2 + 2x - 1)/(Deltax)`
= `lim_(Deltax -> 0) (2xDeltax + (Deltax)^2 - 2Deltax)/(Deltax)`
= `lim_(Deltax -> 0) (Deltax[2x + Deltax - 2])/(Deltax)`
= `lim_(Deltax -> 0) [2x + Deltax - 2]` ...[∵ Δx → 0 ∴ Δx ≠ 0]
= 2x + 0 – 2
= 2x – 2
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