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Question
If f(x) `{:(= (5^x + 5^(-x) - 2)/(x^2)"," , "for" x ≠ 0),(= k",", "for" x = 0):}}` is continuous at x = 0, find k
Solution
f(0) = k ...(Given) ...(1)
`lim_(x -> 0) "f"(x) = lim_(x -> 0) (5^x + 5^(-x) - 2)/(x^2)`
= `lim_(x -> 0) (5^x(5^x + 5^(-x) - 2))/(5^x * x^2)`
= `lim_(x -> 0) ((5^x)^2 + 1 - 2*5^x)/(5^x * x^2)`
= `lim_(x -> 0) (5^x - 1)^2/(5^x * x^2)`
= ` lim_(x -> 0) ((5^x - 1)/x)^2 1/5^x`
= `(lim_(x -> 0) (5^x - 1)/x)^2 xx 1/(lim_(x -> 0) 5^x`
= `(log5)^2 xx 1/5^0 ...[because lim_(x -> 0) ("a"^x - 1)/x = log "a"]`
= (log 5)2 ...(2)
Since f is continuous at x = 0,
f(0) = `lim_(x -> 0) "f"(x)`
∴∴k = (log 5)2 ...[By (1) and (2)]
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