Question

f(x) = `{{:((1 - cos "k"x)/(xsinx)",",   "if"  x ≠ 0),(1/2",",  "if"  x = 0):}` at x = 0

Answer 1

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