Çözüldü Trigonometrik Soldan Limit ve Özdeşlikler

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1 + sin(x) = [ cos(x / 2) + sin(x / 2) ]^2
[ 1 + sin(x) ]^0,5 = { [ cos(x / 2) + sin(x / 2) ]^2 }^0,5
[ 1 + sin(x) ]^0,5 = | [ cos(x / 2) + sin(x / 2) ] |
x < 0 için | [ cos(x / 2) + sin(x / 2) ] | = -[ cos(x / 2) + sin(x / 2) ]....(I)
cosx = [ cos(x / 2) ]^2 - [ sin(x / 2) ]^2 = [ cos(x / 2) + sin(x / 2) ][ cos(x / 2) - sin(x / 2) ]....(II)
(I) ve (II) eşitlikleri limiti aranan ifadedeki yerlerine yazılıp;
lim x → (-π / 2)^(-) 4[ cos(x / 2) + sin(x / 2) ][ cos(x / 2) - sin(x / 2) ] / { -[ cos(x / 2) + sin(x / 2) ] } =
-lim x → (-π / 2)^(-) 4[ cos(x / 2) - sin(x / 2) ] = -4[ cos(-π / 4) - sin(-π / 4) ] =
-4[ cos(π / 4) + sin(π / 4) ] = -4[ (√2) / 2 + (√2) / 2 ] = -4√2

WolframAlpha Kontrolu: https://www.wolframalpha.com/input/?i=lim 4cos(x)/sqrt(1+sin(x)) as x goes to -pi/2 from left