Limits and Derivatives

Limits and Derivatives

Question

Evaluate space space space space space space limit as straight x rightwards arrow 0 of straight f left parenthesis straight x right parenthesis comma space where space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction comma end cell cell straight x not equal to 0 end cell row cell 0 space comma end cell cell straight x equals 0 end cell end table close

Answer

L.H.L    =  space space space space space limit as straight x rightwards arrow 0 to the power of minus of space straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 0 to the power of minus of fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction                                                           [∵ here x < 0]

      space space space space space space space space space space space equals space limit as straight x rightwards arrow 0 to the power of minus of straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 0 to the power of minus of left parenthesis negative 1 right parenthesis equals negative 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space for space straight x space less than 0 comma space open vertical bar x close vertical bar equals negative x right square bracket

R.H.L.     space space space space equals space limit as straight x rightwards arrow 0 to the power of plus space of straight f left parenthesis straight x right parenthesis space equals limit as straight x rightwards arrow 0 to the power of plus of fraction numerator open curly brackets straight x close curly brackets over denominator straight x end fraction space space space space space space space space space space space space space space space space space space space space                                          [∵  here x > 0]

      
              space space space space space equals space limit as straight x rightwards arrow 0 to the power of plus of straight x over straight x                                                                 [ ∵ for x >0, space space space space space open vertical bar straight x close vertical bar equals straight x]

           space space space space space space space space space space space space space equals space space limit as straight x rightwards arrow 0 to the power of plus of left parenthesis 1 right parenthesis equals 1
Thus,               L.H.L  space space not equal to    R.H.L.
∴  space space space limit as straight x rightwards arrow 0 of straight f left parenthesis straight x right parenthesis space does not exist.

More Chapters from Limits and Derivatives