Limits and Derivatives

Limits and Derivatives

Question

Solve   space space space space space space space space space space limit as straight z rightwards arrow 1 of fraction numerator straight z to the power of begin display style 1 third end style end exponent minus 1 over denominator straight z to the power of begin display style 1 over 6 end style end exponent minus 1 end fraction

Answer
space space space space space space space space space space space space space space limit as straight z rightwards arrow 1 of fraction numerator straight z to the power of begin display style 1 third end style end exponent minus 1 over denominator straight z to the power of begin display style 1 over 6 end style minus 1 end exponent end fraction equals limit as straight z rightwards arrow 1 of fraction numerator open parentheses straight z begin display style fraction numerator 1 third minus 1 over denominator straight z minus 1 end fraction end style close parentheses over denominator open parentheses straight z fraction numerator 1 over 6 minus 1 over denominator straight z minus 1 end fraction close parentheses end fraction equals fraction numerator limit as straight z rightwards arrow 1 of begin display style fraction numerator straight z to the power of begin display style 1 third end style end exponent minus 1 over denominator straight z minus 1 end fraction end style over denominator limit as straight z rightwards arrow 1 of fraction numerator straight z to the power of begin display style 1 over 6 end style end exponent minus 1 over denominator straight z minus 1 end fraction end fraction equals fraction numerator begin display style 1 third end style left parenthesis 1 right parenthesis to the power of begin display style 1 third end style minus 1 end exponent over denominator begin display style 1 over 6 end style left parenthesis 1 right parenthesis to the power of begin display style 1 over 6 end style minus 1 end exponent end fraction equals 1 third straight x 6 over 1 equals 2
                                                                                      (By dividing Num and den by z -1)

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