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Question
Find f(a), if f is continuous at x = a where,
f(x) = `(1 - cos[7(x - pi)])/(5(x - pi)^2`, for x ≠ π at a = π
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Solution
f is continuous at x = π
∴ f(π) = `lim_(x -> pi) "f"(x) = lim_(x -> pi) (1 - cos[7 (x - pi)])/(5(x - pi)^2`
Put x – π = h, as x → π, h → 0
∴ f(π) = `lim_("h" -> 0) (1 - cos7"h")/(5"h"^2)`
= `lim_("h" -> 0) (2sin^2((7"h")/2))/(5"h"^2)`
= `2/5 lim_("h" -> 0) (sin^2 ((7"h")/2))/(((7"h")/2)^2) xx (7/2)^2`
= `2/5 |lim_("h" -> 0) (sin ((7"h")/2))/(((7"h")/2))|^2 xx 49/4`
= `2/5 xx (1)^2 xx 49/4 ...[because lim_(theta -> 0) sintheta/theta = 1]`
∴ f(π) = `49/10`
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