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show that the function f(x)=|x+2|is continuous but fails to differentiable at x=-2
Lt      f(x) =Lt    -(x-2)
 x?-2         x?-2
               =0
f(-2) = 0
So f(x) is continuous at x = -2.

Since f(x) = |x+2| = -(x+2)  if x ? -2
 
                          = x+2   if x > -2     
 
       ,
and since we know that both -(x+2) and x+2 are differentiable, the only point where something can go wrong is when x = -2. At this point, we can compute the limit of the difference quotient directly:

    lim
    h?0       f(-2+h) - f(-2)
                 -------------------
                         h
            =     lim      |h|
                    h?0    -------
                               h  
this limit does not exist .



                            
 
        .


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