Limits And Continuity

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The limit of f(x) as x approaches a is l, written as lim[x→a] f(x) = l, if: 1. f(x) is defined in some neighborhood of a (but not necessarily at a) 2. As x gets arbitrarily close to a from both sides…

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Step 1: Direct substitution gives 0/0 (indeterminate form) Step 2: Factor the numerator: x² - 9 = (x - 3)(x + 3) Step 3: Simplify: (x² - 9)/(x - 3) = (x - 3)(x + 3)/(x - 3) = x + 3 (for x ≠ 3) Step 4:…

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Left-hand limit (LHL): lim[x→a⁻] f(x) - approach a from values less than a Right-hand limit (RHL): lim[x→a⁺] f(x) - approach a from values greater than a Key Point: lim[x→a] f(x) exists only if LHL =…

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Geometric Proof: Step 1: Consider unit circle with angle x (in radians) Step 2: Compare areas: △OAC < sector OAB < △OBD Step 3: Area relationships: (1/2)cos x sin x < x/2 < (1/2)tan x Step 4: Multiply…

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Method 1 - Using trigonometric identity: Step 1: Use identity 1 - cos x = 2sin²(x/2) Step 2: lim[x→0] (2sin²(x/2))/x² Step 3: = lim[x→0] 2 × [sin(x/2)/(x/2)]² × (1/4) Step 4: = 2 × (1)² × (1/4) = 1/2 …

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Step 1: Direct substitution gives 0/0 form Step 2: Rationalize by multiplying by conjugate: = lim[x→0] (√(1+x) - 1)/x × (√(1+x) + 1)/(√(1+x) + 1) Step 3: Numerator becomes: (1+x) - 1 = x Step 4: = lim…

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L'Hôpital's Rule: If lim[x→a] f(x)/g(x) gives 0/0 or ∞/∞ form, then: lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x) Conditions for application: 1. f(x) and g(x) are differentiable near x = a 2. lim[x→a] f…

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Method: Divide numerator and denominator by highest power of x (x²) Step 1: = lim[x→∞] (3x²/x² + 2x/x² + 1/x²)/(2x²/x² - x/x² + 5/x²) Step 2: = lim[x→∞] (3 + 2/x + 1/x²)/(2 - 1/x + 5/x²) Step 3: As x…