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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY2 Marks
Question
If $\text{f(x)}=\begin{cases}\frac{1-\cos\text{x}}{\text{x}^2},&\text{x}\neq0\\\text{k},&\text{x}=0\end{cases}$ is continuous at x = 0, find k.
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Answer

Since f(x) is continuous at x = 0,
$\text{f}(0)=\lim\limits_{{\text{x}}\rightarrow0}\text{f(x})$
$\Rightarrow\text{k}=\lim\limits_{{\text{x}}\rightarrow0}\frac{1-\cos\text{x}}{\text{x}^2}$
$\Rightarrow\text{k}=\lim\limits_{{\text{x}}\rightarrow0}\frac{2\sin^2\frac{\text{x}}{2}}{\text{x}^2}$
$\Rightarrow\text{k}=\frac{1}{2}$

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