Limits, Continuity & Differentiability
Derivatives from First Principles
Grade 12
Question:
<p>\(\lim_{h \to 0} \frac{f(2h + 2 + h^2) - f(2)}{f(h + h^2 + 1) - f(1)}\), given that <span class="math inline">\(f'(2) = 6\)</span> and <span class="math inline">\(f'(1) = 4\)</span> [2003 AIEEE]</p>
<p>(a) <span class="math inline">\(\frac{3}{2}\)</span></p>
<p>(b) Does not exist</p>
<p>(c) Is equal to <span class="math inline">\(-\frac{3}{2}\)</span></p>
<p>(d) Is equal to 3</p>
Step-by-Step Solution
Key Concept: Use the definition of derivative to convert the limit into a ratio of derivatives.
<p>Rewrite the numerator and denominator using the definition of derivative. As <span class="math inline">$h \to 0$</span>: numerator <span class="math inline">$\approx f(2 + h) - f(2) \approx f'(2) \cdot h = 6h$</span> and denominator <span class="math inline">$\approx f(1 + h) - f(1) \approx f'(1) \cdot h = 4h$</span>. Therefore, <span class="math inline">$\lim_{h \to 0} \frac{6h}{4h} = \frac{6}{4} = \frac{3}{2}$</span></p>
Correct Answer: A