Limits, Continuity & Differentiability
Existence of Limits using One-sided Limits
Grade 12
Question:
<p>Find <span class="math">\lim_{x \to 2} \frac{x - 2}{|x - 2|}\</span></p>
<p>(a) 1</p>
<p>(b) –1</p>
<p>(c) 0</p>
<p>(d) does not exist</p>
Step-by-Step Solution
Key Concept: A limit exists at a point only if the left-hand limit and right-hand limit are equal. When they differ, the limit does not exist.
<p><strong>Step 1:</strong> Express the absolute value function in piecewise form:</p><p>\[|x - 2| = \begin{cases} x - 2, & \text{if } x \geq 2 \\ -(x - 2), & \text{if } x < 2 \end{cases}\]</p><p><strong>Step 2:</strong> Find the Right Hand Limit (RHL):</p><p>\[\text{RHL} = \lim_{x \to 2^+} \frac{x - 2}{x - 2} = \lim_{x \to 2^+} 1 = 1\]</p><p><strong>Step 3:</strong> Find the Left Hand Limit (LHL):</p><p>\[\text{LHL} = \lim_{x \to 2^-} \frac{x - 2}{-(x - 2)} = -\lim_{x \to 2^-} \frac{x - 2}{x - 2} = -1\]</p><p><strong>Step 4:</strong> Compare LHL and RHL:</p><p>Since \(\lim_{x \to 2^+} f(x) = 1 \neq -1 = \lim_{x \to 2^-} f(x)\), the left and right hand limits are not equal.</p><p>∴ The limit does not exist. Answer is (d).</p>
Correct Answer: D