Limits — Flashcards
ICSE · Class 11 · Mathematics
31 flashcards for Limits (ICSE Class 11 Mathematics) to test yourself on key terms and facts. Sample: "Solve: lim_{x→3}(x+3)"
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Solve: lim_{x→3}(x+3)
Answer
Step 1: Use direct substitution because the expression is defined at x = 3. Step 2: Substitute x = 3. lim_{x→3}(x+3) = 3 + 3 Step 3: Simplify. Answer: 6…
Solve: lim_{x→1}[(x^2+1)/(x+100)]
Answer
Step 1: Use direct substitution because the function is defined at x = 1. Step 2: Substitute x = 1. lim_{x→1}[(x^2+1)/(x+100)] = [(1)^2+1]/[1+100] Step 3: Simplify. = 2/101 Answer: 2/101…
Solve: lim_{x→a}[(x-a)/(x+a)], where a ≠ 0
Answer
Step 1: Use direct substitution since x = a is allowed if the denominator is not zero. Step 2: Substitute x = a. [(a-a)/(a+a)] = 0/(2a) Step 3: Simplify. Answer: 0…
When do you use the factorisation method?
Answer
Use factorisation when direct substitution gives the indeterminate form 0/0. Example: lim_{x→3}[(x^2-9)/(x-3)] Step 1: Factor the numerator: x^2-9 = (x-3)(x+3) Step 2: Cancel the common factor (x-3).
Solve: lim_{x→3}[(x^2-9)/(x-3)]
Answer
Step 1: Direct substitution gives 0/0, so factorise. Step 2: x^2-9 = (x-3)(x+3) Step 3: Cancel (x-3). lim_{x→3}[(x^2-9)/(x-3)] = lim_{x→3}[x+3] Step 4: Substitute x = 3. = 6 Answer: 6…
Solve: lim_{x→2}[(x^3-2x^2)/(x^2-5x+6)]
Answer
Step 1: Factor numerator and denominator. x^3-2x^2 = x^2(x-2) x^2-5x+6 = (x-2)(x-3) Step 2: Cancel the common factor (x-2). lim_{x→2}[x^2/(x-3)] Step 3: Substitute x = 2. = 4/(2-3) Step 4: Simplif…
Solve: lim_{x→1/2}[(8x-3)/(2x-1) - (4x^2+1)/(4x^2-1)]
Answer
Step 1: Write 4x^2-1 = (2x-1)(2x+1). Step 2: Take common denominator. [(8x-3)(2x+1) - (4x^2+1)] / [(2x-1)(2x+1)] Step 3: Simplify the numerator. (8x-3)(2x+1) - (4x^2+1) = 12x^2+2x-4 Step 4: Factor. 12…
When do you use the rationalisation method?
Answer
Use rationalisation when square roots are present and direct substitution gives 0/0. Example: lim_{x→0}[x/(sqrt(a+x)-sqrt(a-x))] Step 1: Multiply numerator and denominator by the conjugate sqrt(a+x)+s…
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