[QUIZ] Berapa nilai limitnya.... Question from @Diradiradira

November 2019 1 91 Report

[QUIZ] Berapa nilai limitnya

$\lim_{h \to 0} \frac{tan(h-x)-tan(-h-x)}{h\sqrt{4-h^2}}=$

$\lim \limits_{h \to 0} \frac{\tan(h-x)-\tan(-h-x)}{h\sqrt{4-h^2}} \\ = \lim \limits_{h \to 0} \frac{1}{\sqrt{4-h^2}} \cdot \frac{\tan(h-x)+\tan(h+x)}{h} \\ = \frac{1}{\sqrt{4-(0)^2}}\cdot \lim \limits_{h \to 0} \frac{\frac{\sin(h-x)}{\cos(h-x)}+\frac{\sin(h+x)}{\cos(h+x)}}{h} \\ = \frac{1}{2} \cdot \lim \limits_{h \to 0} \frac{\frac{\sin(h-x)\cos(h+x)+\sin(h+x)\cos(h-x)}{\cos(h-x)\cos(h+x)}}{h} \\ = \frac{1}{2} \cdot \frac{1}{\cos\big((0)-x\big)\cos\big((0)+x\big)} \cdot \lim \limits_{h \to 0} \frac{\sin(h-x)\cos(h+x)+\sin(h+x)\cos(h-x)}{h} \\ = \frac{1}{2 \cos^{2} (x) } \cdot \lim \limits_{h \to 0} \frac{\sin(h-x)\cos(h+x)+\sin(h+x)\cos(h-x)}{h} \\ = \frac{ \sec ^{2} (x) }{2} \cdot \lim \limits_{h \to 0} \frac{\sin(h-x)\cos(h+x)+\sin(h+x)\cos(h-x)}{h}$

Aturan L'Hopital:

$= \frac{ \sec ^2(x)}{2} \cdot \lim \limits_{h \to 0} \frac{\frac{d}{dh} \big(\sin(h-x)\cos(h+x)+\sin(h+x)\cos(h-x)\big)}{\frac{dh}{dh}}$

Turunan dari sin(h-x) cos(h+x) + sin(h+x) cos(h-x) terhadap h:

$\frac{d}{dh} \big(\sin(h-x)\cos(h+x)+\sin(h+x)\cos(h-x)\big)\\ = \big(\cos(h-x)\cos(h+x)-\sin(h-x)\sin(h+x) \big)+\big(\cos(h+x)\cos(h-x)-\sin(h+x)\sin(h-x) \big)\\ = 2\big(\cos(h-x)\cos(h+x)-\sin(h+x)\sin(h-x) \big)\\ = 2\cos\big((h-x)+(h+x) \big)\\ = 2\cos(2h)$

Sehingga:

$= \frac{ \sec ^2(x)}{2} \cdot \lim \limits_{h \to 0} \frac{\frac{d}{dh} \big(\sin(h-x)\cos(h+x)+\sin(h+x)\cos(h-x)\big)}{\frac{dh}{dh}}\\ = \frac{\sec^2(x)}{2} \cdot \lim \limits_{h \to 0} \frac{2\cos(2h)}{1}\\ = \sec^2(x) \cdot \cos\big(2(0)\big)\\ = \sec^2(x)$

Jawaban:

sec²(x)

MRizky099 cos(-x) = cos(x) → cos(-x) cos(x) = cos²(x). Maaf saya menjawab sepertinya banyak diloncat-loncat

MRizky099 cos(a) cos(b) - sin(a) sin(b) = cos(a+b)

diradiradira nice, tapi sepertinya kalo menggunakan sifat tan(a+b) = (tana +tanb)/(1-tanatanb) bisa sedikit lebih praktis

diradiradira good job anyway

MRizky099 Saya hanya cari aman saja. Soalnya lupa sifat penjumlahan tan. Btw, makasih dah mengingatkan dan saya coba lagi pakai sifat tan justru jadi bingung