Limits, Continuity & Differentiability
Continuity and Discontinuity
Grade 12
Question:
<p>Let <br>
\[f(x) = \begin{cases} \dfrac{1}{\ln x} & \text{if } x > 0,\ x \neq 1 \\ \dfrac{1}{\ln(-x)} & \text{if } x < 0,\ x \neq -1 \end{cases}\]<br>
The number of points of discontinuity of \(f(x)\) is:</p>
<p>(1) 1</p>
<p>(2) 2</p>
<p>(3) 3</p>
<p>(4) 4</p>
Step-by-Step Solution
Key Concept: A function is discontinuous where it's undefined, where the limit doesn't exist, or where the limit doesn't equal the function value. Here, analyze domain restrictions (ln x undefined at x=1 and x=-1) and behavior at x=0 separately for each piece.
<p><strong>Step 1: Identify domain restrictions.</strong></p><p>For <em>x > 0, x ≠ 1:</em> f(x) = 1/ln(x) is undefined at x = 1 (ln 1 = 0).</p><p>For <em>x < 0:</em> f(x) = 1/ln(-x) is undefined at x = -1 (ln 1 = 0).</p><p><strong>Step 2: Check continuity at x = 1.</strong></p><p>lim(x→1⁺) 1/ln(x) and lim(x→1⁻) don't exist (limits → ±∞). <strong>Discontinuous at x = 1.</strong></p><p><strong>Step 3: Check continuity at x = -1.</strong></p><p>lim(x→-1⁻) 1/ln(-x) and lim(x→-1⁺) don't exist (limits → ±∞). <strong>Discontinuous at x = -1.</strong></p><p><strong>Step 4: Check continuity at x = 0.</strong></p><p>lim(x→0⁺) 1/ln(x) → -∞ and lim(x→0⁻) 1/ln(-x) → -∞, but f(0) is not defined in either piece. <strong>Discontinuous at x = 0.</strong></p><p><strong>Step 5: Check continuity elsewhere.</strong></p><p>Both pieces are continuous on their respective domains (R+ ∖ {1} and R⁻ ∖ {-1}).</p><p>∴ <strong>Answer: (3) 3 points of discontinuity at x = -1, 0, and 1</strong></p>
Correct Answer: (3) 3