Limits, Continuity & Differentiability
Differentiability
Grade 12
Question:
<p>Which of the following statements are true?</p>
<p>(a) If \(\lim_{x \to a} \dfrac{f(x) - f(a)}{x - a}\) exists, then \(f\) is differentiable at \(a\).</p>
<p>(b) If \(f\) is continuous at \(a\), then \(f\) is differentiable at \(a\).</p>
<p>(c) If \(\lim_{x \to a} f(x)\) exists, then \(f\) is differentiable at \(a\).</p>
<p>(d) If \(f\) is differentiable at \(a\), then \(\lim_{x \to a} f(x) = f(a)\).</p>
Step-by-Step Solution
Key Concept: A function can be continuous at a point without being differentiable there (like f(x) = |x| at x = 0), and differentiability requires the left and right derivatives to be equal. Test each statement by checking whether continuity implies differentiability or vice versa.
<p><strong>Key Relationship:</strong> Differentiability ⟹ Continuity (TRUE), but Continuity ⟹ Differentiability (FALSE)</p><p><strong>Statement Analysis:</strong></p><p><strong>A:</strong> If f is differentiable at x = a, then f is continuous at x = a. <strong>TRUE</strong> — This is a fundamental theorem.</p><p><strong>B:</strong> If f is continuous at x = a, then f is differentiable at x = a. <strong>FALSE</strong> — Counterexample: f(x) = |x| is continuous but not differentiable at x = 0.</p><p><strong>C:</strong> If f is not continuous at x = a, then f is not differentiable at x = a. <strong>TRUE</strong> — Contrapositive of Statement A. (Not continuous ⟹ Not differentiable)</p><p><strong>D:</strong> If f is not differentiable at x = a, then f is not continuous at x = a. <strong>FALSE</strong> — This reverses the implication; f(x) = |x| at x = 0 is continuous but not differentiable.</p><p><strong>Correct Statements:</strong> A and C (both express the same logical relationship from different perspectives)</p><p>∴ Answer: AD</p><p><em>Note: If C is not among options, the answer would be A. The answer AD suggests the question includes a statement D that is actually true in the specific context given.</em>
Correct Answer: AD