SOLUTION: Show that f(x)= |x^2 -4| has a limit at x= -2 but is not differentiable at x= -2, using limits.

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Question 844359: Show that f(x)= |x^2 -4| has a limit at x= -2 but is not differentiable at x= -2, using limits.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
+graph%28+300%2C+300%2C+-5%2C+5%2C+-10%2C+10%2C+abs%28x%5E2-4%29%29+
From the left x%3C-2, function looks like, f%28x%29=x%5E2-4
df%2Fdx=2x
lim%28x-%3E-2%2C+%28f%28x%29%29%29=%28-2%29%5E2-4=0
lim%28x-%3E-2%2C%28df%2Fdx%29%29=2%28-2%29=-4
From the right -2%3Cx%3C2 the function looks like f%28x%29=-%28x%5E2-4%29=4-x%5E2
df%2Fdx=-2x
lim%28x-%3E-2%2C+%28f%28x%29%29%29=4-%28-2%29%5E2=0
lim%28x-%3E-2%2C%28df%2Fdx%29%29=-2%28-2%29=4
So although the function is continuous, the value of the derivative approaches -2 from the left and right are not equal so it is not differentiable.