document.write( "Question 844359: Show that f(x)= |x^2 -4| has a limit at x= -2 but is not differentiable at x= -2, using limits. \n" ); document.write( "

Algebra.Com's Answer #509089 by Fombitz(32388) $\"\"$ $\"About$

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$\"+graph%28+300%2C+300%2C+-5%2C+5%2C+-10%2C+10%2C+abs%28x%5E2-4%29%29+\"$
\n" ); document.write( "From the left $\"x%3C-2\"$ , function looks like, $\"f%28x%29=x%5E2-4\"$
\n" ); document.write( " $\"df%2Fdx=2x\"$
\n" ); document.write( " $\"lim%28x-%3E-2%2C+%28f%28x%29%29%29=%28-2%29%5E2-4=0\"$
\n" ); document.write( " $\"lim%28x-%3E-2%2C%28df%2Fdx%29%29=2%28-2%29=-4\"$
\n" ); document.write( "From the right $\"-2%3Cx%3C2\"$ the function looks like $\"f%28x%29=-%28x%5E2-4%29=4-x%5E2\"$
\n" ); document.write( " $\"df%2Fdx=-2x\"$
\n" ); document.write( " $\"lim%28x-%3E-2%2C+%28f%28x%29%29%29=4-%28-2%29%5E2=0\"$
\n" ); document.write( " $\"lim%28x-%3E-2%2C%28df%2Fdx%29%29=-2%28-2%29=4\"$
\n" ); document.write( "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. \n" ); document.write( "

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