Limits and Derivatives

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A derivative represents the rate of change of a function at a specific point. For example, if distance s = 4.9t², the derivative at t = 2 gives the instantaneous velocity of the object at that moment.

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The limit of function f(x) as x approaches 'a' is the value that f(x) gets arbitrarily close to as x gets closer to 'a'. Notation: lim(x→a) f(x) = L, read as 'limit of f(x) as x tends to a equals L'.

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Left-hand limit: lim(x→a⁻) f(x) - value approached when x approaches 'a' from left. Right-hand limit: lim(x→a⁺) f(x) - value approached when x approaches 'a' from right. A limit exists only when both

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If lim(x→a) f(x) and lim(x→a) g(x) both exist, then: • lim(x→a) [f(x) + g(x)] = lim(x→a) f(x) + lim(x→a) g(x) • lim(x→a) [f(x) - g(x)] = lim(x→a) f(x) - lim(x→a) g(x) The limit of sum/difference equal

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If lim(x→a) f(x) and lim(x→a) g(x) both exist, then: • Product: lim(x→a) [f(x)·g(x)] = lim(x→a) f(x) × lim(x→a) g(x) • Quotient: lim(x→a) [f(x)/g(x)] = lim(x→a) f(x) / lim(x→a) g(x), provided lim(x→a)

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For polynomial f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ: lim(x→a) f(x) = f(a) = a₀ + a₁a + a₂a² + ... + aₙaⁿ Simply substitute the value of 'a' into the polynomial. This is because polynomials are continuo

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lim(x→a) (xⁿ - aⁿ)/(x - a) = naⁿ⁻¹ Proof: (xⁿ - aⁿ) = (x - a)(xⁿ⁻¹ + xⁿ⁻²a + ... + xaⁿ⁻² + aⁿ⁻¹) So the limit = aⁿ⁻¹ + aⁿ⁻¹ + ... + aⁿ⁻¹ (n terms) = naⁿ⁻¹ This formula is fundamental for finding der

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1. lim(x→0) (sin x)/x = 1 2. lim(x→0) (1 - cos x)/x = 0 These are fundamental limits used to find derivatives of trigonometric functions. The first limit can be proved using the squeeze theorem with