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General
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Question

L t subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator sec space x minus tan space x over denominator pi over 2 minus x end fraction

  1. 2
  2. 1 half
  3. 1
  4. pi over 2

Hint:

To simplify the equation we will use some trigonometric identities and put straight pi over 2 minus x space equals y then we will again simplify using trigonometric identities.

The correct answer is: 1 half


    L t subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator sec space x minus tan space x over denominator pi over 2 minus x end fraction
    equals limit as x rightwards arrow straight pi over 2 of fraction numerator begin display style fraction numerator 1 over denominator cos space x end fraction end style minus begin display style fraction numerator sin space x over denominator cos x end fraction end style over denominator begin display style straight pi over 2 end style minus x end fraction
equals limit as x rightwards arrow straight pi over 2 of open parentheses fraction numerator 1 minus sin space x over denominator cos space x end fraction close parentheses open parentheses fraction numerator 1 over denominator begin display style straight pi over 2 end style minus x end fraction close parentheses
straight pi over 2 minus x equals y rightwards double arrow x equals straight pi over 2 minus y
equals limit as y rightwards arrow 0 of fraction numerator 1 minus sin open parentheses begin display style straight pi over 2 end style minus y close parentheses over denominator y cos open parentheses begin display style straight pi over 2 end style minus y close parentheses end fraction
equals limit as y rightwards arrow 0 of fraction numerator 1 minus cos space y over denominator y. sin y end fraction
equals limit as y rightwards arrow 0 of fraction numerator 1 minus cos y over denominator y squared end fraction. fraction numerator y over denominator sin space y end fraction
equals limit as y rightwards arrow 0 of fraction numerator 2 sin squared begin display style y over 2 end style over denominator open parentheses begin display style y over 2 end style close parentheses squared 4 end fraction. fraction numerator y over denominator sin y end fraction
equals 2 over 4 equals 1 half

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