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Question
f(x) = `{{:((1 - cos "k"x)/(xsinx)",", "if" x ≠ 0),(1/2",", "if" x = 0):}` at x = 0
Solution
We have f(x) = `{{:((1 - cos "k"x)/(xsinx)",", "if" x ≠ 0),(1/2",", "if" x = 0):}`
Since, f(x) is continuous at x = 0
∴ f(0) = `lim_(x -> 0) "f"(x)`
∴ `1/2 = lim_(x -> 0) (1 - cos "k"x)/(xsinx)`
= `lim_(x -> 0) (1 - cos^2"k"x)/(xsinx) * 1/(1 + cos "k"x)`
= `lim_(x -> 0) (sin^2"k"x)/(xsinx) * 1/(1 + cos"k"x)`
= `lim_(x -> 0) (((sin "k"x)/("k"x))^2 "k"^2)/((sinx)/x) * 1/(1 + cos "k"x)`
= `"k"^2/1 * 1/(1 + 1)`
= `"k"^2/2`
⇒ k2 – 1
⇒ k = ±1
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