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Limits and Derivatives

Short Answer Type

21.

Evaluate $space space limit as straight x rightwards arrow straight x over 4 of fraction numerator cos space straight x space minus space sin space straight x over denominator straight x minus straight pi divided by 4 end fraction.$

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22.

Evaluate $space space space space space limit as straight x rightwards arrow straight pi of fraction numerator sin left parenthesis straight n minus straight pi right parenthesis over denominator straight pi left parenthesis straight pi minus straight x right parenthesis end fraction.$

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23.
Evaluate $space space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space space ax plus bx over denominator ax plus sin space bx end fraction comma space straight a comma space space straight b comma space straight a plus straight b not equal to 0$

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 fraction numerator sin space ax plus bx over denominator ax plus sin space bx end fraction equals limit as straight x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin space ax space plus space space bx over denominator straight x end fraction end style over denominator begin display style fraction numerator ax plus sin space space bx over denominator straight x end fraction end style end fraction equals fraction numerator begin display style fraction numerator sin space ax over denominator straight x end fraction plus straight b end style over denominator straight a plus begin display style fraction numerator sin space bx over denominator straight x end fraction end style end fraction

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 space space space space space space space space space space space space equals space limit as straight x rightwards arrow 0 of fraction numerator begin display style fraction numerator straight a space sin space ax over denominator ax end fraction plus straight b end style over denominator straight a plus begin display style fraction numerator straight b space sin space bx over denominator bx end fraction end style end fraction equals fraction numerator straight a. limit as ax rightwards arrow 0 of begin display style fraction numerator siin space ax over denominator ax end fraction end style plus straight b over denominator straight a plus straight b. limit as bx rightwards arrow 0 of begin display style fraction numerator sin space bx over denominator bx end fraction end style end fraction equals fraction numerator straight a plus straight b over denominator straight a plus straight b end fraction equals 1

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24.

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