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Question
Find the value of k for which the function f given as
f(x) =`{{:((1 - cosx)/(2x^2)",", if x ≠ 0),( k",", if x = 0 ):}`
is continuous at x = 0.
Solution
f(x) = `{{:((1 - cosx)/(2x^2)",", if x ≠ 0),( k",", if x = 0 ):}` is continous at x = 0
= `lim_(x rightarrow 0) (2 sin^2 x/2)/(2 x^2/4 . 4)` ...`(∵ lim_(x rightarrow 0) sinx/x = 1)`
= `lim_(x rightarrow 0) ((sin x/2)/(x/2))^2 . 1/4`
So k = `1/4`
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