Limit x Tends to Zero $\lim_{x\rightarrow 0}\frac{1 – cosx}{x^2}$

(i) Limit x Tends to Zero $\lim_{x\rightarrow 0}\frac{1 – cosx}{x^2}$

= $\lim_{x\rightarrow 0}\frac{\cos^2\frac{x}{2} + sin^2\frac{x}{2} – \cos^2\frac{x}{2} + sin^2\frac{x}{2}}{x^2}$

= $\lim_{x\rightarrow 0}\frac{2sin^2\frac{x}{2}}{x^2}$

=$ \lim_{x\rightarrow 0}\frac{2sin^2\frac{x}{2}}{2\left(\frac{x}{2}\right)^2}$

= $\frac{1}{2}\lim_{x\rightarrow 0}\frac{2sin^2\frac{x}{2}}{\left(\frac{x}{2}\right)^2}$

= $\frac{1}{2}\lim_{x\rightarrow 0}\left(\frac{\sin \frac{x}{2}}{\frac{x}{2}}\right)^2$

= $\frac{1}{2}\ldotp 1$

= $\frac{1}{2}$