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Maths-

General

Easy

Question

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

2
$\frac{1}{2}$
1
$\frac{π}{2}$

hint 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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