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Question
Determine whether \[f\left( x \right) = \binom{\frac{\sin x^2}{x}, x \neq 0}{0, x = 0}\] is continuous at x = 0 or not.
Solution
Given:
\[\lim_{x \to 0} f\left( x \right) = \lim_{x \to 0} \frac{\sin x^2}{x}\]
\[ = \lim_{x \to 0} \frac{x \sin x^2}{x^2}\]
\[ = \lim_{x \to 0} \frac{\sin x^2}{x^2} \lim_{x \to 0} x\]
\[ = 1 \times 0\]
\[ = 0\]
\[ = f\left( 0 \right)\]
\[\therefore \lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]
Hence
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