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Question
Find f(a), if f is continuous at x = a where,
f(x) = `(1 + cos(pi x))/(pi(1 - x)^2)`, for x ≠ 1 and at a = 1
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Solution
f(x) is continuous at x = 1
∴ f(1) = `lim_(x -> 1) "f"(x)`
∴ f(1) = `lim_(x -> 1) (1 + cos pix)/(pi(1 - x)^2`
Put 1 – x = h
∴ x = 1 – h
As x → 1, h → 0
∴ f(1) = `lim_("h" -> 0) (1 + cos[pi(1 - "h")])/(pi"h"^2)`
= `lim_("h" -> 0) (1 + cos(pi - pi"h"))/(pi"h"^2)`
= `lim_("h" -> 0) (1 - cos pi"h")/(pi"h"^2)`
= `lim_("h" -> 0) (1 - cos pi"h")/(pi"h"^2) xx (1 + cos pi"h")/(1 + cos pi"h")`
= `lim_("h" -> 0) (1 - cos^2 pi"h")/(pi"h"^2 (1 + cos pi"h")`
= `1/pi lim_("h" -> 0) (sin^2 pi"h")/("h"^2 (1 + cos pi"h"))`
= `1/pi lim_("h" -> 0) ((sin pi"h")/"h")^2 xx 1/(1 + cos pi"h")`
= `1/pi lim_("h" -> 0) ((sin pi"h")/(pi"h"))^2 xx pi^2 xx 1/(lim_("h" -> 0) (1 + cos pi"h"))`
= `1/pi xx (1)^2 xx pi^2 xx 1/(1 + 1) ...[("As" "h" -> 0"," pi"h" -> 0),("and" lim_(theta -> 0) sintheta/theta = 1)]`
= `pi/2`
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