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Question
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) `{:( = (sin^2pix)/(3(1 - x)^2) ",", "for" x ≠ 1),(= (pi^2sin^2((pix)/2))/(3 + 4cos^2 ((pix)/2)) ",", "for" x = 1):}}` at x = 1
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Solution
f(1) = `(pi^2sin^2 (pi/2))/(3 + 4cos^2 (pi/2))`
= `(pi^2 xx 1^2)/(3 + 0)`
= `pi^2/3` ...(1)
`lim_(x -> 1) "f"(x) = lim_(x -> 1) (sin^2pix)/(3(1 - x)^2`
= `lim_(x -> 1) (sin^2(pi - pix))/(3(1 - x)^2` ...[∵ sin (π – θ) = sin θ]
= `lim_(x -> 1) [sin{pi(1 - x)}]^2/(3pi^2(1 - x)^2) * pi^2`
= `pi^2/3 [lim_(x -> 1) (sin{pi(1 - x)})/[pi(1 - x))]^2`
= `pi^2/3 xx 1^2 ...[(because x -> 1"," (x - 1) -> 0 "," therefore pi(1 - x) -> 0),("and" lim_(theta -> 0) sintheta/theta = 1)]`
= `pi^2/3` ...(2)
From (1) and (2),
`lim_(x -> 1) "f"(x)` = f(1)
∴ f is continuous at x = 1
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