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प्रश्न
If f is continuous at x = 0 then find f(0) where f(x) = `[5^x + 5^-x - 2]/x^2`, x ≠ 0
उत्तर
f is continuous at x = 0
∴ `lim_(x -> 0) f(x) = f(0)`
∴ `lim_(x -> 0) [5^x + 5^-x - 2]/x^2 = f(0)`
∴ `f(0) = lim_(x -> 0) [5^x + 1/5^x - 2]/x^2`
= `lim_( x -> 0) [(5^x)^2 + 1 - 2(5^x)]/x^2 xx 1/5^x`
= `lim_( x -> 0) (5^x - 1)^2/[x^2] xx 1/5^x`
= `lim_( x -> 0) ((5^x - 1)/x)^2 xx lim_( x -> 0)(1/5^x)`
= (log5)2 x `1/5^0`
= (log5)2 x 1
∴ f(0) = (log5)2
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