Question

Find k if the following function is continuous at the points indicated against them:

`f(x) = (1 + kx)^(1/x)`  , for x ≠ 0

       = `e^(3/2)`                  ,  for x = 0, at x = 0

Answer 1

f is continuous at x = 0

∴ `lim_(x→0) "f"(x)` = f(0)

∴ `lim_(x→0) (1 + "k"x)^(1/x) = "e"^(3/2)`

∴ `lim_(x→0) [(1 + "k"x)^(1/("k'"x)]]^"k" = "e"^(3/2)`

∴ `"e"^"k" = "e"^(3/2) ....[lim_(x→0) (1 + x)^(1/x) = "e"]`

∴ k = `3/2`