Question

The value of $\displaystyle \lim_{x \to 0} \frac{\sqrt{1-cos\,x^{2}}}{1-cos\,x}$ is

• A. $\frac{1}{2}$
• B. $2$
• C. $\sqrt{2}$
• D. None of these

Answer 1

The Correct Option is C

$ \displaystyle\lim_{x \to 0} \frac{\sqrt{1-cos\,x^{2}}}{1-cos\,x} = \displaystyle\lim _{x \to 0} \frac{\sqrt{2\, sin^{2} \frac{x^{2}}{2}}}{2\, sin^{2} \frac{x}{2}}$ $= \displaystyle\lim _{x \to 0} \frac{\sqrt{2} \left|sin \frac{x^{2}}{2}\right|}{2\, sin^{2} \frac{x}{2}}$ $LHL = \displaystyle\lim _{x \to 0^{-}} \frac{\sqrt{2} \left|sin \frac{x^{2}}{2}\right|}{2 \,sin^{2} \frac{x}{2}} = \displaystyle\lim _{h \to 0} \frac{\sqrt{2} \left|sin \frac{\left(-h\right)^{2}}{2}\right|}{2\, sin^{2} \frac{\left(-h\right)}{2}}$ $\displaystyle\lim _{h \to 0} \frac{\sqrt{2}\, sin \frac{h^{2}}{2}}{2 \,sin^{2} \frac{h}{2}}= \frac{\sqrt{2}}{2} \displaystyle\lim _{h \to 0} \frac{ sin\, \frac{h^{2}}{2}}{ \frac{h^{2}}{2}}\times\frac{\left(\frac{h}{2}\right)^{2}}{\left(sin \frac{h}{2}\right)^{2}}\times2$ [Note that the question contains mod sign, hence we checked for LHL and RHL]