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 space space space space space space space space space space space space space space space limit as straight x rightwards arrow 0 of left parenthesis cosec space straight x minus cot space straight x right parenthesis

Answer
space space space space 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 left parenthesis cosec space straight x space minus space cot space straight x right parenthesis space equals space limit as straight x rightwards arrow 0 of open parentheses fraction numerator 1 over denominator siin space straight x end fraction minus fraction numerator cos space straight x over denominator siin space straight x end fraction close parentheses equals limit as straight x rightwards arrow 0 of fraction numerator 1 minus cos space straight x over denominator sin space straight x end fraction equals limit as straight x rightwards arrow 0 of fraction numerator 2 sin squared begin display style straight x over 2 end style over denominator sin space straight x end fraction

      space space space space space space space space space space space space space space space space equals space 2 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 sin space straight x end fraction equals limit as straight x rightwards arrow of fraction numerator begin display style fraction numerator sinx divided by 2 over denominator straight x divided by 2 end fraction end style over denominator begin display style fraction numerator sin space straight x over denominator straight x end fraction end style end fraction. sin straight x over 2

            space space space space space space space space space space space space space equals space fraction numerator limit as begin display style straight x over 2 end style rightwards arrow 0 of begin display style fraction numerator sinx divided by 2 over denominator straight x divided by 2 end fraction end style over denominator limit as straight x rightwards arrow 0 of begin display style fraction numerator sin space straight x over denominator straight x end fraction end style end fraction. limit as straight x rightwards arrow 0 of sin straight x over 2 equals 1 over 1.0 space equals 0

More Chapters from Limits and Derivatives