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Question
If f(x) = `(4^(x - π) + 4^(π - x) - 2)/(x - π)^2` for x ≠ π, is continuous at x = π, then find f(π).
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Solution
f is given to be continuous at x = π
∴ by definition,
f(π) = `lim_(x -> pi) "f"(x)`
= `lim_(x -> pi) (4^(x - pi) + 4^(pi - x) - 2)/(x - pi)^2`
Put x = π + h. Then as x → π, h → 0 and x – π = h
∴ f(π) = `lim_("h" -> 0) (4^"h" + 4^(-"h") - 2)/"h"^2`
= `lim_("h" -> 0) (4^"h"[4^"h" + 4^(-"h") - 2])/(4^"h" * "h"^2)`
= `lim_("h" -> 0) ((4^"h")^2 + 1 - 2(4^"h"))/(4^"h" * "h"^2)`
= `lim_("h" -> 0) (4^"h" - 1)^2/(4^"h" * "h"^2)`
= `[lim_("h" -> 0) (4^"h" - 1)/"h"]^2/(lim_("h" -> 0) 4^"h")`
= `(log 4)^2/4^0 ...[lim_(x -> 0) ("a"^x - 1)/x = log "a"]`
= (2 log 2)2
∴ f(π) = 4(log 2)2
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