Limits, Continuity & Differentiability
Standard Limits
Grade 12
Question:
<p>If <span class="math">f(x) = \begin{cases} \frac{\log(1+2ax)-\log(1+bx)}{x} & , x \neq 0 \\ k & , x = 0 \end{cases}</span> is continuous at <span class="math">x=0</span>, then <span class="math">k</span> is equal to</p>
<p>(a) <span class="math">2a-b</span></p>
<p>(b) <span class="math">2a+b</span></p>
<p>(c) <span class="math">b-2a</span></p>
<p>(d) <span class="math">a+b</span></p>
Step-by-Step Solution
Key Concept: Apply the standard logarithmic limit formula by decomposing and factoring appropriately.
<p><strong>Solution:</strong> Use the standard limit <span class="math">\lim_{x \to 0} \frac{\log(1+u)}{u} = 1</span>. <span class="math">\lim_{x \to 0} \frac{\log(1+2ax)-\log(1+bx)}{x} = \lim_{x \to 0} \left[\frac{\log(1+2ax)}{x} - \frac{\log(1+bx)}{x}\right] = \lim_{x \to 0} \left[2a\frac{\log(1+2ax)}{2ax} - b\frac{\log(1+bx)}{bx}\right] = 2a(1) - b(1) = 2a-b</span>. Therefore <span class="math">k = 2a-b</span>.</p>
Correct Answer: a