Question

If the function f is continuous at x = 0 then find f(0),
where f(x) =  `[ cos 3x - cos x ]/x^2`, `x!=0`

Answer 1

Consider

`lim_( x -> 0) f(x) = lim_( x -> 0) [ (cos3x - cos x)/x^2 ]`

                           = `lim_( x -> 0) [ [2sin2x sin(-x)]/x^2 ]`

                           = `-2 lim_( x -> 0) ((sin 2x)/x). lim_( x -> 0)((sin x)/x)`

                           = `-2 lim_( x -> 0) ((sin 2x)/2x). 2 . lim_( x -> 0)((sin x)/x)`

                           = -2 . 2

                          `( ∵ lim_( x -> 0 ) sin x/x = 1 and as  x → 0, 2x → 0 )`
                          = -4
Since f is continuous at x = 0
                          = `lim_( x -> 0) f(x) = F(0)`
                     -4  = f(0)
                ∴  f(0) = -4