Question

If f is continuous at x = 0 then find f(0) where f(x) = `[5^x + 5^-x - 2]/x^2`, x ≠ 0

Answer 1

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