Question

If  \[f\left( x \right) = x \sin\frac{1}{x}, x \neq 0,\]then the value of the function at x = 0, so that the function is continuous at x = 0, is

 

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Answer 1

Given: 

\[f\left( x \right) = x \sin\frac{1}{x}, x \neq 0\]

Here,

\[\lim_{x \to 0} x \sin \left( \frac{1}{x} \right) = \lim_{x \to 0} x \lim_{x \to 0} \sin\left( \frac{1}{x} \right) = 0 \times \lim_{x \to 0} \sin\left( \frac{1}{x} \right) = 0\]

If f(x) is continuous at x = 0, then

\[\lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]

\[\Rightarrow f\left( 0 \right) = 0\]