<p>\(\lim_{x \to 2} [x]\) exists where \([x]\) denotes the integral part of \(x\).</p><p><em>State whether the statement is true or false.</em></p>
Step-by-Step Solution
Key Concept: The limit of a function exists at a point only if the left-hand limit equals the right-hand limit. For the greatest integer function [x], the left and right limits at integer values are different, so the limit doesn't exist at x=2.
<p><strong>Step 1:</strong> Find the left-hand limit as x approaches 2 from the left (x→2⁻).</p><p>For values slightly less than 2 (like 1.9, 1.99, 1.999...), we have 1 < x < 2, so [x] = 1.</p><p>Therefore, lim(x→2⁻)[x] = 1</p><p><strong>Step 2:</strong> Find the right-hand limit as x approaches 2 from the right (x→2⁺).</p><p>For values slightly greater than 2 (like 2.1, 2.01, 2.001...), we have 2 ≤ x < 3, so [x] = 2.</p><p>Therefore, lim(x→2⁺)[x] = 2</p><p><strong>Step 3:</strong> Compare the left and right limits.</p><p>Since lim(x→2⁻)[x] = 1 ≠ 2 = lim(x→2⁺)[x], the left-hand limit does not equal the right-hand limit.</p><p><strong>Step 4:</strong> Conclusion.</p><p>For a limit to exist at a point, both one-sided limits must be equal. Since they are unequal here, lim(x→2)[x] does <strong>not exist</strong>.</p><p>∴ The statement is <strong>False</strong></p>
Correct Answer: B