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Question
Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :
f(x) = `(1 - cos2x)/sinx`, for x ≠ 0
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Solution
f(x) = `(1 - cos2x)/sinx`, for x ≠ 0
Here, f(0) is not defined.
Consider,
`lim_(x -> 0) "f"(x) = lim_(x -> 0) (1 - cos 2x)/sinx`
= `lim_(x -> 0) (2sin^2x)/sinx`
= `2 lim_(x -> 0) (sin x) ...[(because x -> 0"," therefore x ≠ 0),(therefore sin x ≠ 0)]`
= 2(sin 0)
= 2 × 0
= 0.
∴ `lim_(x -> 0) "f"(x)` exists.
But f(0) is not been defined.
∴ f(x) has a removable discontinuity at x = 0.
∴ The extension of the original function is
f(x) = `{:((1 - cos 2x)/sinx ,";" "for" x ≠ 0),(= 0, ";" "for" x = 0):}`
∴ f(x) is continuous at x = 0
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