Limits, Continuity & Differentiability
Evaluation of Limits
Grade 12
Question:
<p>If \(\lim_{x \to 1}\left(2 - x + a[x-1] + b[1+x]\right)\) exists, then <em>a</em> and <em>b</em> can take values (where [.] denotes greatest integer function)</p>
<p>(a) \(a = 1/3, b = 1\)</p>
<p>(b) \(a = 1, b = -1\)</p>
<p>(c) \(a = 9, b = -9\)</p>
<p>(d) \(a = 2, b = 2/3\)</p>
Step-by-Step Solution
Key Concept: For a limit to exist at x=1 involving greatest integer functions, the left and right limits must be equal; this requires the greatest integer function terms to have compatible behavior as x approaches 1 from both sides.
<p><strong>Step 1:</strong> Analyze the function as x → 1⁻ and x → 1⁺</p><p><strong>Step 2 (Left limit, x → 1⁻):</strong> When x → 1⁻, we have x < 1, so [x-1] = -1 and [1+x] = 1 (since 1 < 1+x < 2)<br/>∴ L⁻ = (2 - 1) + a(-1) + b(1) = 1 - a + b</p><p><strong>Step 3 (Right limit, x → 1⁺):</strong> When x → 1⁺, we have x > 1, so [x-1] = 0 and [1+x] = 2 (since 2 < 1+x < 3)<br/>∴ L⁺ = (2 - 1) + a(0) + b(2) = 1 + 2b</p><p><strong>Step 4:</strong> For limit to exist: L⁻ = L⁺<br/>1 - a + b = 1 + 2b<br/>-a + b = 2b<br/>-a = b<br/>∴ <strong>a = -b</strong> or equivalently <strong>a + b = 0</strong></p><p>∴ Answer: C</p>
Correct Answer: C