Question

Question 21.
Find the value(s) of k so that the following function is continuous at x = 0 :
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Answer 1

We have to find the value of k if the given function is continuous at x = 0.

Now,

$\rm \large\displaystyle\lim_{ \rm x \to0}{\rm \: f(x)} =\rm f(0)$

$\rm\longrightarrow\displaystyle\lim_{ \rm x \to0}{\rm \: f(x)} = \displaystyle\lim_{ \rm x \to0}{\rm \: \dfrac{1 - cos \: kx}{x \: sin \: x} } \\ \\ \\ \rm\longrightarrow \displaystyle\lim_{ \rm x \to0}{\rm \: \dfrac{ \dfrac{(1 - cos \: kx) \times {(kx)}^{2} }{ {(kx)}^{2} } }{ {x}^{2} \: \bigg(\dfrac{sin \: x}{x} \bigg) } } \\ \\ \\ \rm\longrightarrow \displaystyle\lim_{ \rm x \to0}{\rm \: \dfrac{ \dfrac{1 }{ {2} } \times {(kx)}^{2} }{ {x}^{2} } }\\ \\ \\ \rm\longrightarrow \displaystyle {\dfrac{ \rm {k}^{2} }{ 2 } }...(1)$

$\rm f(0) = \dfrac{1}{2} ...(2)$

Now, from (1) and (2) :-

$\rm \rightarrow \dfrac{ {k}^{2} }{2} = \dfrac{1}{2} \\ \\ \\\rm \rightarrow { {k}^{2} } = 1\\ \\ \\ \rm \rightarrow { {k} } = \pm 1$