Limits help!!!

Question

  
Determine whether the function is continuous or discontinous. Justify your reasoning. If discontinuous, identify points of discontinuity.

 

 

              2-X^2 if X <= 1

ƒ(x)= {------------------------

              X if X>1

 

Answer 1

as every algebraic function is continuous in its domain so f(x) is continuous for x<1 and x>1 but as the definition of f(x) is different on the either side of the point x=1. So we need to check the continuity at x=1 only.

 

For that we will find Left hand limit and right hand limit ..if both comes same then f(x) will be continuous everywhere if not then discontinuous at x=1.

 

 

LHL at x=1 :    lim f(1-h)= lim    ( 2-(1-h)²)  =lim   (2-(1)²+2h-h²) =1

                         h->0         h->0                     h->0

 

RHL at x=1 :    lim f(1+h)= lim   (1+h )     = 1

                         h->0           h->0    

 

       As,        LHL =RHL ; So f(x) is continuous