Limits

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It means that as x approaches the value 'a' (but never equals 'a'), the function f(x) approaches the value L. In other words, the limiting value of f(x) is L when x tends to a. Key point: x ≠ a, so (x…

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Step 1: Check if direct substitution is possible (denominator ≠ 0) Step 2: Since this is a polynomial, substitute x = 3 directly Step 3: lim(x→3) (2x + 5) = 2(3) + 5 = 6 + 5 = 11 Answer: 11…

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A limit does not exist at x = a when: 1. Left-hand limit ≠ Right-hand limit (lim(x→a⁻) f(x) ≠ lim(x→a⁺) f(x)) 2. The function approaches ±∞ 3. The function oscillates without approaching a specific va…

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Step 1: Direct substitution gives 0/0 (indeterminate form) Step 2: Factor the numerator: x² - 4 = (x + 2)(x - 2) Step 3: lim(x→2) (x² - 4)/(x - 2) = lim(x→2) [(x + 2)(x - 2)]/(x - 2) Step 4: Cancel (x…

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If lim(x→a) f(x) = L and lim(x→a) g(x) = M, then: lim(x→a) [f(x) ± g(x)] = lim(x→a) f(x) ± lim(x→a) g(x) = L ± M This means the limit of a sum/difference equals the sum/difference of individual limi…

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Step 1: Multiply by conjugate: [√(1+x) + √(1-x)]/[√(1+x) + √(1-x)] Step 2: lim(x→0) [(1+x) - (1-x)]/[x(√(1+x) + √(1-x))] Step 3: Simplify numerator: (1+x) - (1-x) = 2x Step 4: lim(x→0) 2x/[x(√(1+x) + …

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lim(x→0) (aˣ - 1)/x = log a (where a > 0) Special case: lim(x→0) (eˣ - 1)/x = log e = 1 This is a fundamental limit used for exponential functions. Remember: the base 'a' must be positive.

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Step 1: Recognize this fits the pattern lim(x→a) (xⁿ - aⁿ)/(x - a) = n(aⁿ⁻¹) Step 2: Here a = 1, n = 3 Step 3: Apply formula: lim(x→1) (x³ - 1³)/(x - 1) = 3(1³⁻¹) = 3(1²) = 3(1) = 3 Alternatively by …