Limits and Derivatives — Practice

Question 1

Evaluate: lim(x→2) (x²−4)/(x−2).

Accepted Answer

Direct substitution gives 0/0. Factor: (x²−4) = (x−2)(x+2). So (x²−4)/(x−2) = (x+2) for x≠2. lim(x→2) (x+2) = 2+2 = 4.

Question 2

Find the derivative of f(x) = x³ from first principles.

Accepted Answer

f'(x) = lim(h→0) [f(x+h)−f(x)]/h = lim(h→0) [(x+h)³−x³]/h. Expand: (x+h)³ = x³+3x²h+3xh²+h³. So (x+h)³−x³ = 3x²h+3xh²+h³. Divide by h: 3x²+3xh+h². Take limit h→0: f'(x) = 3x².

Question 3

Differentiate y = x² sin x using the product rule, then evaluate dy/dx at x = π/2.

Accepted Answer

Using product rule with u=x², v=sinx: dy/dx = u'v+uv' = 2x·sinx + x²·cosx = 2x sinx + x² cosx. At x=π/2: dy/dx = 2(π/2)sin(π/2) + (π/2)²cos(π/2) = π×1 + (π²/4)×0 = π.

Practice — Limits and Derivatives

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