Question:

<p><strong>Step 1:</strong> Rewrite the limit condition:</p><p>$$\lim_{t \to x} \frac{\sec x \cdot f(t) - f(x) \sec t}{t - 1} = \sec^2 x$$</p><p><strong>Step 2:</strong> This limit must hold for all x. Rearrange as a derivative:</p><p>The numerator suggests: $$\frac{d}{dt}[\sec x \cdot f(t) - f(x) \sec t]\bigg|_{t=x} = \sec x \cdot f'(x) - f(x) \sec x \tan x$$</p><p><strong>Step 3:</strong> Since the limit equals $\sec^2 x$, we have:</p><p>$$\sec x \cdot f'(x) - f(x) \sec x \tan x = \sec^2 x$$</p><p><strong>Step 4:</strong> Divide by $\sec x$:</p><p>$$f'(x) - f(x) \tan x = \sec x$$</p><p><strong>Step 5:</strong> Evaluate at $x = 0$ where $f(0) = 1$, $\tan 0 = 0$, $\sec 0 = 1$:</p><p>$$f'(0) - 1 \cdot 0 = 1$$</p><p>$$f'(0) = 1$$</p><p><strong>Step 6:</strong> Calculate the required ratio:</p><p>$$\frac{f(0)}{f'(0)} = \frac{1}{1} = 1$$</p><p>∴ Answer: C (or 1)</p>