Limits and Derivatives

Limits and Derivatives

Question

Evaluate space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator 1 minus cos space straight x over denominator straight x end fraction.

Answer
space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator 1 minus cos space straight x over denominator straight x end fraction equals limit as straight x rightwards arrow 0 of fraction numerator 2 sin squared straight x divided by 2 over denominator straight x end fraction equals limit as straight x rightwards arrow 0 of fraction numerator sin begin display style straight x over 2 end style. sin begin display style straight x over 2 end style over denominator begin display style straight x over 2 end style end fraction
       space space space space space space space space space space space space space space space equals space limit as straight x rightwards arrow 0 of fraction numerator sinx divided by 2 over denominator straight x divided by 2 end fraction straight x limit as straight x rightwards arrow 0 of sin straight x over 2 equals 1 straight x 0 equals 0

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