Word Count : 503

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Let's go through each statement one by one. ### (i) \(\lim_{x \to 0} \frac{x^2 \sin\left(\frac{1}{x}\right)}{\sin x} = 1\) False. To evaluate this limit, we first note that: - \(\sin\left(\frac{1}{x}\right)\) oscillates between -1 and 1 as \(x \to 0\). - Thus, \(x^2 \sin\left(\frac{1}{x}\right)\) is bounded between \(-x^2\) and \(x^2\), which goes to 0 as \(x \to 0\). - \(\sin x \approx x\) for small \(x\), so \(\frac{x^2 \sin\left(\frac{1}{x}\right)}{\sin x} \approx \frac{x^2}{x} = x \to 0\). Therefore, the limit does not equal 1; it approaches 0. ### (ii) A real-valued function of three variables which is continuous everywhere is differentiable. False. Continuity alone does not guarantee differentiability. A famous counterexample is the function \(f(x, y, z) = \left\{\begin{array}{ll} \frac{x^2 + y^2 + z^2}{x^2 + y^2 + z^2} _______ ____ ___ _______ ________ __________ ______ ____ _________ ________ ______.
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