Limits, Continuity & Differentiability
Multiple limits and their evaluation
Grade 12
Question:
<p>\(\lim_{t \to 0} \lim_{x \to \infty} \cot\left(\frac{\pi(1 - t^2f(x) \cdot g(x))}{4}\right)\) is equal to</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) \(e\)</p>
<p>(d) does not exist</p>
Step-by-Step Solution
Key Concept: Evaluate the double limit by first determining the behavior of f(x)·g(x) as x → ∞, then apply the cotangent function.
<p>As \(x \to \infty\), \(f(x) \cdot g(x)\) tends to a limit that makes the argument of cotangent approach \(\frac{\pi}{4}\), so \(\cot\left(\frac{\pi}{4}\right) = 1\). Taking the limit as \(t \to 0\) preserves this value.</p>
Correct Answer: b