Limits, Continuity & Differentiability
Limits
Grade 12
Question:
<p>\(\lim_{x \to 1} f(x)\) exists if \(f(x)\) is defined as follows:<br>\(f(x) = x^2, x < 1\)<br>\(= x, x = 1\)<br>\(= x^2 + 2x, x > 1\)</p><p><em>State whether the statements are true or false.</em></p>
<p>(a) True</p>
<p>(b) False</p>
Step-by-Step Solution
Key Concept: For a limit to exist at a point, the left-hand limit and right-hand limit must be equal. You must evaluate lim(x→1⁻) and lim(x→1⁺) separately using the appropriate piece of the function definition.
<p><strong>Step 1: Identify the function pieces at x = 1</strong></p><p>For x < 1: f(x) = x²</p><p>At x = 1: f(1) = 1</p><p>For x > 1: f(x) = x² + 2x</p><p><strong>Step 2: Calculate Left-Hand Limit (LHL)</strong></p><p>lim(x→1⁻) f(x) = lim(x→1⁻) x² = 1² = 1</p><p><strong>Step 3: Calculate Right-Hand Limit (RHL)</strong></p><p>lim(x→1⁺) f(x) = lim(x→1⁺) (x² + 2x) = 1² + 2(1) = 1 + 2 = 3</p><p><strong>Step 4: Check if limit exists</strong></p><p>Since LHL = 1 and RHL = 3, we have LHL ≠ RHL</p><p>Therefore, lim(x→1) f(x) <strong>does NOT exist</strong></p><p><strong>Statement is FALSE</strong></p><p>∴ Answer: B (False)</p>
Correct Answer: B