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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`
Solution
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
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