Question

If  \[f\left( x \right) = \binom{\frac{1 - \cos x}{x^2}, x \neq 0}{k, x = 0}\]  is continuous at x = 0, find k. 

Answer 1

Given: 

\[f\left( x \right) = \binom{\frac{1 - \cos x}{x^2}, x \neq 0}{k, x = 0}\]

If  \[f\left( x \right)\]  is continuous at  \[x = 0\] , then

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

\[\Rightarrow \lim_{x \to 0} \left( \frac{1 - \cos x}{x^2} \right) = k\]
\[ \Rightarrow \lim_{x \to 0} \left( \frac{2 \left[ \sin \left( \frac{x}{2} \right) \right]^2}{4 \left( \frac{x}{2} \right)^2} \right) = k\]
\[ \Rightarrow \frac{1}{2} \lim_{x \to 0} \left( \frac{\left[ \sin\left( \frac{x}{2} \right) \right]^2}{\left( \frac{x}{2} \right)^2} \right) = k\]
\[ \Rightarrow 1 \times \frac{1}{2} = k\]
\[ \Rightarrow k = \frac{1}{2}\]