Question

Discuss the continuity of the function f at x = 0

If f(x) = ${\displaystyle \frac{2^{3x}-1}{\tan x}}$, for x ≠ 0

         = 1,   for x = 0

Answer 1

Given f(0) = 1

Consider,

${\displaystyle \underset{x\to 0}{lim}}$ f (x) = ${\displaystyle \underset{x\to 0}{lim}\left[\frac{2^{3x}-1}{\tan x}\right]}$

  = ${\displaystyle \underset{x\to 0}{lim}\left[\frac{\frac{2^{3x}-1}{x}}{\frac{\tan x}{x}}\right],x\ne 0}$

= ${\displaystyle \underset{x\to 0}{lim}\frac{\frac{2^{3x}-1}{3x}.3}{\underset{x\to 0}{lim}\frac{\tan x}{x}}=3\log 2}$

= log 8

${\displaystyle (\underset{x\to 0}{lim}\frac{a^x-1}{x}=\log a\text{and}\underset{x\to 0}{lim}\frac{\tan x}{x}=1)}$

${\displaystyle asx\to 0,3x\to 0}$

Since ${\displaystyle (\underset{x\to 0}{lim}}$ f(x) ≠ f(0)

f(x) is discontinuous at x = 0