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

Answer 1

Function f is continuous at x = 0

∴ `"f"(0) = lim_(x→0) "f"(x)`

∴ k = `lim_(x→0) (5^x + 5^-x - 2)/(x^2)`

= `lim_(x→0) (5^x + 1/5^x - 2)/(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)`  ...[∵ a² - 2ab + b² = (a - b)²]

= `lim_(x→0) ((5^x - 1)/x)^2. 1/5^x`

= `lim_(x→0) ((5^x - 1)/x)^2 xx lim_(x→0) 1/5^x`

= `(log 5)^2 xx 1/5^0   ....[∵ lim_(x→0)(("a"^x - 1)/x) = log "a"]`

∴ k = (log 5)²