Limits, Continuity & Differentiability
Limits and logical implications
Grade 12
Question:
<p>Consider the following statements about positive functions \(f(x)\) and \(g(x)\) whose limits to infinity exist.</p><p>Statement-A: \(\lim_{x\to\infty} f(x) = \lim_{x\to\infty} g(x)\)</p><p>Statement-B: \(\lim_{x\to\infty}(f(x)-g(x))=0\)</p><p>Statement-C: \(\lim_{x\to\infty}\sqrt{f(x)} = \lim_{x\to\infty}\sqrt{g(x)}\)</p><p>How many of the following six statements are true: \(A\Rightarrow B,\; B\Rightarrow C,\; C\Rightarrow A,\; A\Rightarrow C,\; B\Rightarrow A,\; C\Rightarrow B\)?</p>
Step-by-Step Solution
Key Concept: Test each implication by checking if the hypothesis logically forces the conclusion using limit properties; A and C are equivalent (both imply equal limits of f and g), while B is weaker and only implies C, not A.
<p><strong>Analysis of each implication:</strong></p><p><strong>A⇒B:</strong> If lim f(x) = lim g(x) = L, then lim(f(x)-g(x)) = L - L = 0. ✓ TRUE</p><p><strong>B⇒C:</strong> If lim(f(x)-g(x)) = 0, then f(x) and g(x) are asymptotically equal. Thus lim√f(x) = lim√g(x). ✓ TRUE</p><p><strong>C⇒A:</strong> If lim√f(x) = lim√g(x) = M, then lim f(x) = M² = lim g(x). ✓ TRUE</p><p><strong>A⇒C:</strong> If lim f(x) = lim g(x) = L, then lim√f(x) = √L = lim√g(x) (by continuity of √·). ✓ TRUE</p><p><strong>B⇒A:</strong> Counterexample: f(x) = x, g(x) = x+1. Then lim(f-g) = -1 ≠ 0, so this fails. But if we modify: f(x)=x, g(x)=x+1/x gives lim(f-g)=0 but lim f ≠ lim g. ✗ FALSE</p><p><strong>C⇒B:</strong> Counterexample: f(x) = x, g(x) = x². Then √f(x) = √x and √g(x) = x both diverge but not equally; actually √f(x)→∞ and √g(x)→∞ differently. More clearly: f(x)=x, g(x)=4x have √f→∞, √g→∞ at same rate, but f-g = -3x → -∞ ≠ 0. ✗ FALSE</p><p><strong>Summary:</strong> A⇒B, B⇒C, C⇒A, A⇒C are TRUE (4 implications)</p><p>∴ Answer: <strong>4</strong></p>
Correct Answer: 4