If $f(x) = \left\{ \begin{array}{r} \frac{1 - \cos x}{x},\ x
eq 0 \ \ k\ ,\ \ \ \ x = 0 \end{array} ight.\$ is continuous at $x = 0$, then the value of $k$ is

Question

If $f(x) = \left\{ \begin{array}{r} \frac{1 - \cos x}{x},\ x 
eq 0 \ \ k\ ,\ \ \ \ x = 0 \end{array} ight.\$ is continuous at $x = 0$, then the value of $k$ is

Dr. Abubacker · Accepted Answer

(a)
$\lim_{x \rightarrow 0}{f(x) =}\lim_{x \rightarrow 0}\ \frac{1 - \cos x}{x} = \lim_{x \rightarrow 0}\frac{2\sin^{2}\frac{x}{2}}{4\left( \frac{x}{2} \right)^{2}}.x = 0$
Also, $f(0) = k$
For, $\lim_{x \rightarrow 0}\ \ f(x) = f(0)\ \ \  \Rightarrow \ k = 0$

Q. If $f(x) = \left\{ \begin{array}{r} \frac{1 - \cos x}{x},\ x \neq 0 \\ \ k\ ,\ \ \ \ x = 0 \end{array} \right.\$ is continuous at $x = 0$, then the value of $k$ is

Explanation

Reviewed by Dr. Abubacker

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