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Question Number 121052 by mathocean1 last updated on 05/Nov/20

$$\mathrm{show}\mathrm{that}$$$$\underset{\mathrm{x}\to 0}{\lim }\frac{1-\mathrm{cosx}}{{\mathrm{x}}^2}=\frac{1}{2}$$$$$$

Answered by Dwaipayan Shikari last updated on 05/Nov/20

$$\underset{x\to 0}{\lim }\frac{1-cosx}{x^2}=\underset{x\to 0}{\lim }\frac{2{sin}^2\frac{x}{2}}{x^2}=\underset{x\to 0}{\lim }\frac{2x^2}{4x^2}=\frac{1}{2}$$

Answered by malwan last updated on 05/Nov/20

$$\underset{x\to 0}{lim}\frac{2{sin}^2\frac{x}{2}}{x^2}=2{\left(\frac{1}{2}\right)}^2=2\times \frac{1}{4}=\frac{1}{2}$$

Answered by 675480065 last updated on 05/Nov/20

$$\mathrm{x}\to 0,\mathrm{cosx}\to 1-\frac{{\mathrm{x}}^2}{2}+\frac{{\mathrm{x}}^4}{24}+0\left({\mathrm{x}}^4\right)$$$$\Rightarrow \underset{x\to 0}{\lim }\frac{1-\mathrm{cosx}}{{\mathrm{x}}^2}=\underset{x\to 0}{\lim }\frac{1-1+\frac{{\mathrm{x}}^2}{2}-\frac{{\mathrm{x}}^4}{24}}{{\mathrm{x}}^2}=\underset{x\to 0}{\lim }\frac{{\mathrm{x}}^2(\frac{1}{2}-\frac{{\mathrm{x}}^2}{24})}{{\mathrm{x}}^2}$$$$=\underset{x\to 0}{\lim }(\frac{1}{2}-\frac{{\mathrm{x}}^2}{24})=\frac{1}{2}$$

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