Question

Discuss the continuity of the function f at x = 0

If f(x) = `(2^(3x) - 1)/tanx`, for x ≠ 0

         = 1,   for x = 0

Answer 1

Given f(0) = 1

Consider,

`lim_(x->0)` f (x) = `lim_(x->0) [(2^(3x) - 1)/tanx]`

  = `lim_(x->0) [((2^(3x) - 1)/x)/((tanx)/x]], x ≠ 0`

= `lim_(x->0) [(2^(3x) - 1)/(3x).3]/(lim_(x->0)(tanx)/x) = 3 log 2`

= log 8

`(lim_(x->0) (a^x - 1)/x = log a and lim_(x->0) (tan x)/x = 1)`

`as x -> 0, 3x ->0`

Since `(lim_(x->0)` f(x) ≠ f(0)

f(x) is discontinuous at x = 0