



 
Hi Edith, Let's write f(x) = √(x  1). If x  1 < 0 (that is x < 1) then √(x  1) doesn't exist and hence x is not in the domain of the function. Hence f(x) = √(x  1) is not differentiable if x < 1. If x  1 > 0 (that is x > 1) then x is in the domain of the function. To determine that f is differentiable at x you need to evaluate
Notice that since x  1 > 0, if h > 0 then (x  1  h) > 0 and the square root exists. Also if h < 0 then for sufficiently small h, (x  1 + h) > 0 and again the square root exists. Now you need to perform the algebra to determine if the limit exists. The final possibility is x  1 = 0. In this case the limit becomes
In this case when h < 0 the square root doesn't exist and hence the limit can't exist. I hope this helps,  


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