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Continuity And Differentiability

Question
CBSEENMA12034660
Wired Faculty App

Prove space that space straight f left parenthesis straight x right parenthesis equals open vertical bar straight x minus 3 close vertical bar space has space no space derivative space at space straight x equals 3.

Solution
Here f(x) = |x - 3|
straight L. straight H. straight D equals Lt with straight x rightwards arrow 3 to the power of minus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 3 right parenthesis over denominator straight x minus 3 end fraction equals Lt with straight x rightwards arrow 3 to the power of minus below fraction numerator open vertical bar straight x minus 3 close vertical bar minus 0 over denominator straight x minus 3 end fraction space space space space space space left square bracket because space straight f left parenthesis 3 right parenthesis equals open vertical bar 3 minus 3 close vertical bar equals open vertical bar 0 close vertical bar equals 0 right square bracket space space space space space space space space space space space equals Lt with straight x rightwards arrow 3 to the power of minus below fraction numerator open vertical bar straight x minus 3 close vertical bar over denominator straight x minus 3 end fraction space space space space space space space space space space left square bracket Put space straight x equals 3 minus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 3 to the power of minus right square bracket space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar 3 minus straight h minus 3 close vertical bar over denominator 3 minus straight h minus 3 end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar negative straight h close vertical bar over denominator negative straight h end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator straight h over denominator negative straight h end fraction equals negative 1 straight R. straight H. straight D equals Lt with straight x rightwards arrow 3 to the power of plus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 3 right parenthesis over denominator straight x minus 3 end fraction equals Lt with straight x rightwards arrow 3 to the power of plus below fraction numerator open vertical bar straight x minus 3 close vertical bar minus 0 over denominator straight x minus 3 end fraction space space space space space space space space space space space equals Lt with straight x rightwards arrow 3 to the power of plus below fraction numerator open vertical bar straight x minus 3 close vertical bar over denominator straight x minus 3 end fraction space space space space space space space space space left square bracket Put space straight x equals 3 plus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 3 to the power of plus right square bracket space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar 3 plus straight h minus 3 close vertical bar over denominator 3 plus straight h minus 3 end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar straight h close vertical bar over denominator straight h end fraction equals Lt with straight h rightwards arrow 0 below straight h over straight h equals 1 therefore straight L. straight H. straight D not equal to straight R. straight H. straight D therefore space straight f left parenthesis straight x right parenthesis equals open vertical bar straight x minus 3 close vertical bar space is space not space derivable space at space straight x equals 3

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