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Question
Discuss the continuity of f(x) at x = `pi/4` where,
f(x) `{:(= ((sinx + cosx)^3 - 2sqrt(2))/(sin 2x - 1)",", "for" x ≠ pi/4),(= 3/sqrt(2)",", "for" x = pi/4):}`
Solution
`"f"(pi/4) = 3/sqrt(2)`
= `lim_(x -> pi/4) "f"(x) = lim_(x -> pi/4) ((sinx + cosx)^3 - 2sqrt(2))/(sin 2x - 1)`
= (sin x + cos x)3 = `[(sin x + cos x)^2]^(3/2)`
= `(1 + sin 2x)^(3/2)`
∴ `lim_(x -> pi/4) "f"(x) = lim_(x -> pi/4) ((1 + sin 2x)^(3/2) - 2^(3/2))/(sin 2x - 1)`
Put 1 + sin 2x = t
∴ sin 2x = t – 1
As `x -> pi/4"," "t" -> 1 + sin 2(pi/4)`
i.e. `"t" -> 1 + sin pi/(2)`
i.e. t → 1 + 1
i.e. t → 2
∴ `lim_(x -> pi/4) "f"(x) = lim_("t" -> 2) ("t"^(3/2) - 2^(3/2))/("t" - 1 - 1)`
= `lim_("t" -> 2) ("t"^(3/2) - 2^(3/2))/("t" - 2)`
= `3/2(2)^(1/2) ...[lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `(3sqrt(2))/(2)`
= `3/sqrt(2)`
∴ `lim_(x -> pi/4) "f"(x) = "f"(pi/4)`
∴ f(x) is continuous at x = `pi/(4)`
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