Limits, Continuity & Differentiability
Limits and Higher Order Derivatives
Grade 12
Question:
<p>If $\displaystyle \lim_{x \to a} \frac{f(x) - f(a)}{(x-a)^3}$ is a finite non-zero number, then $f(x)$ is of maximum degree</p>
<p>(A) 4</p>
<p>(B) 3</p>
<p>(C) 2</p>
<p>(D) 1</p>
Step-by-Step Solution
Key Concept: Use Taylor series expansion and analyze which terms must vanish for the limit to be finite and non-zero.
<p>For the limit $\displaystyle \lim_{x \to a} \frac{f(x) - f(a)}{(x-a)^3}$ to be finite and non-zero, we use Taylor expansion: $f(x) - f(a) = f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$</p><p>For the limit to be finite, all terms with power less than 3 must vanish (i.e., $f'(a) = f''(a) = 0$). The term $(x-a)^3$ must have a non-zero coefficient, meaning the highest degree is 3.</p>
Correct Answer: B