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प्रश्न
Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :
f(x) = `(3sin^2 x + 2cos x(1 - cos 2x))/(2(1 - cos^2x)`, for x ≠ 0
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उत्तर
f(x) = `(3sin^2 x + 2cos x(1 - cos 2x))/(2(1 - cos^2x)`, for x ≠ 0
∴ f(0) is not defined
`lim_(x -> 0) "f"(x) = lim_(x -> 0) (3sin^2x + 2cosx(1 - cos2x))/(2(1 - cos^x))`
= `lim_(x -> 0) (3sin^2x + 2cosx (2sin^2x))/(2sin^2x)`
= `lim_(x -> 0) (sin^2x (3 + 4cosx))/(2sin^2x)`
= `lim_(x -> 0) (3 + 4cosx)/2 ...[(because x -> 0"," x ≠ 0),(therefore sinx ≠ sin 0 = 0),(therefore sin^2x ≠ 0)]`
= `1/2 lim_(x -> 0) (3 + 4 cos x)`
= `1/2 (3 + 4 cos 0)`
= `7/2`
If we define f(0) = `7/2`, f will be continuous at x = 0
∴ continuous extension = `7/2` i.e. f(0) = `7/2`.
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