We have, Lf′(0)=h→0lim−hf(0−h)−f(0) =h→0lim−hlog(1+h2)−hlogcosh =h→0limlog(1+h2)logcosh(00 form ) =h→0lim2h/(1+h2)−tanh=−1/2 Rf′(0)=h→0limhf(0+h)−f(0) =h→0limhlog(1+h2)hlogcosh =h→0limlog(1+h2)logcosh(00 form ) =h→0lim2h/(1+h2)−tanh=2−1
Since Lf′(0)=Rf;(0), therefore f(x) is differentiable at x=0
Since differentiability ⇒ continuity, therefore f(x) is continuous at x=0.