Question

${lim}_{x\to 0}\frac{\sqrt{\frac{1}{2}(1-\cos x)}}{x}=$

• A. $1$
• B. $-1$
• C. $0$
• D. does not exist

Answer 1

Answer: (D)

does not exist

${lim}_{x\to 0}\frac{\frac{\sqrt{1-cosx}}{2}}{x}$

$={lim}_{x\to 0}\frac{\sqrt{\frac{1}{2}\left(2si{n}^2\frac{x}{2}\right)}}{x}$

$={lim}_{x\to 0}\frac{\left|sin\frac{x}{2}\right|}{x}$

$L.H.L={lim}_{x\to 0}-\frac{-sin\frac{x}{2}}{x}=\frac{-1}{2}$

$R.H.L={lim}_{x\to 0}+\frac{sin\frac{x}{2}}{x}=\frac{1}{2}$

since, $L.H.L\ne R.H.L$

Hence,${lim}_{x\to 0}\frac{\frac{\sqrt{1-cosx}}{2}}{x}$ Doesn't exist.