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.

Answer 1

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`