Derivatives, integrals, limits, and series — with step-by-step workings.
Find the derivative of x²
Using the power rule: d/dx(x²) = 2x.
Evaluate ∫ 2x dx
Integrate term by term: x² + C, where C is the constant of integration.
Find lim(x→0) sin(x)/x
This is a standard limit equal to 1. By the squeeze theorem: for small x ≠ 0, cos(x) ≤ sin(x)/x ≤ 1, and both sides approach 1 as x→0, so lim = 1.
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