Question

if f(x) = { x/tan2x, x>0 , x+2, x≥ 0

find the type of discontinuity and state whether it is continuous

Answer 1

Assume given function as:

f(x) = { x/tan2x, x<0 , x+2, x≥ 0

Solution:

$LHL=\lim_{x\rightarrow0^-}\dfrac x{\tan 2x} \\=\lim_{h\rightarrow0}\dfrac {0-h}{\tan 2(0-h)} \\=\lim_{h\rightarrow0}\dfrac {-h}{-\tan 2h} \\=\lim_{h\rightarrow0}\dfrac {h}{\tan 2h} \\=\lim_{h\rightarrow0}\dfrac {1}{\dfrac{\tan 2h}{2h}\times 2} \\=\dfrac {1}{1\times 2} [\because \lim_{x\rightarrow0}\dfrac{\tan x}{x}=1] \\=\dfrac 12$

$RHL=\lim_{x\rightarrow0^+} (x+2) \\=\lim_{h\rightarrow0} (0+h+2) \\=\lim_{h\rightarrow0}\ 2 \\=2$

$\because LHL\ne RHL$

$\\\therefore$ Given function is discontinuous at $x=0$ which is jump discontinuity.