Question

If $f(x)=\{\begin{array}{cc}\frac{1-\cos x}{x},& x\ne 0\\ k,& x=0\end{array}$ is continuous at $x=0,$ then the value of $k$ is

• A. $0$
• B. $1/2$
• C. $\frac{1}{4}$
• D. $-\frac{1}{2}$
• E. None of these

Answer 1

Answer: (A)

Given, $f(x)=\{\begin{array}{cc}\frac{1-\cos x}{x},& x\ne 0\\ k,& x=0\end{array}$ $\underset{x\to 0}{\lim }f(x)=\underset{x\to 0}{\lim }\frac{1-\cos x}{x}$
$=\underset{x\to 0}{\lim }\frac{2{\sin }^2\frac{x}{2}}{4\left(\frac{x}{2}\right)}.x=0$ and $f(0)=k$
$\because$ Function is continuous at $x=0$ .
$\therefore$ $\underset{x\to 0}{\lim }f(x)=f(0)\Rightarrow k=0$