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

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

space space space space space space space space Put space straight x space equals straight pi plus straight h comma space straight x rightwards arrow straight pi rightwards double arrow straight h rightwards arrow 0

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$space space space space space equals space 1 over straight pi straight x 1 equals 1 over straight pi$

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23.

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

119 Views

24.

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