Limits, Continuity & Differentiability
Limits of piecewise functions
Grade 12
Question:
<p>Find <span class="latex-inline">\lim_{x \to 0^-} f(x)</span> and <span class="latex-inline">\lim_{x \to 1^-} f(x)</span>, where <span class="latex-inline">f(x) = \begin{cases} 2x + 3, & x \leq 0 \\ 3(x+1), & x > 0 \end{cases}</span></p>
<p>(a) <span class="latex-inline">3, 5</span></p>
<p>(b) <span class="latex-inline">3, 6</span></p>
<p>(c) <span class="latex-inline">4, 7</span></p>
<p>(d) <span class="latex-inline">3, -6</span></p>
Step-by-Step Solution
Key Concept: For piecewise functions, identify which piece applies at each limit point
<p><strong>Step 1:</strong> For <span class="latex-inline">\lim_{x \to 0^-} f(x)</span>: Since <span class="latex-inline">x \leq 0</span> near <span class="latex-inline">0^-</span>, use <span class="latex-inline">f(x) = 2x + 3</span></p><p><span class="latex-inline">\lim_{x \to 0^-} f(x) = 2(0) + 3 = 3</span></p><p><strong>Step 2:</strong> For <span class="latex-inline">\lim_{x \to 1^-} f(x)</span>: Since <span class="latex-inline">x > 0</span> for values approaching 1 from left, use <span class="latex-inline">f(x) = 3(x+1)</span></p><p><span class="latex-inline">\lim_{x \to 1^-} f(x) = 3(1 + 1) = 6</span></p><p>∴ Answer is (b) <span class="latex-inline">3, 6</span></p>
Correct Answer: B